Impulse and Linear Momentum 5

Implse and Linea Momentm 5 How does jet poplsion wok? How can yo mease the speed of a bllet? Wold a meteoite collision significantly change Eath s obit? In pevios chaptes we discoveed that the pshing inteaction between ca ties and the oad allows a ca to change its velocity. Likewise, a ship s popelles psh wate backwad; in tn, wate pshes the ship fowad. Bt how does a ocket, fa above Eath s atmosphee, change velocity with no object to psh against? Less than 1 yeas ago, ocket flight was consideed impossible. When U. S. ocket pionee Robet Goddad pblished an aticle Be se yo know how to: Constct a foce diagam fo an object (Section 2.1). Use Newton s second law in component fom (Section 3.2). Use kinematics to descibe an object s motion (Section 1.7). 151

152 Chapte 5 Implse and Linea Momentm in 192 abot ockety and even sggested a ocket flight to the Moon, he was idicled by the pess. A New Yok Times editoial dismissed his idea, saying,... even a schoolboy knows that ockets cannot fly in space becase a vacm is devoid of anything to psh on. We know now that Goddad was coect bt why? What does the ocket psh on? Fige 5.1 The mass is the same in the closed flask (an isolated system) (a) befoe bning the steel wool and (b) afte bning the steel wool. (c) Howeve, the mass inceases (a) when the steel wool is bned in the open flask (a non-isolated system). (a) The block Closed flask (a) The balances block the Closed Steel wool flask balances steel wool the The block Closed Steel wool flask steel and flask. Balancing wool balances and flask. the Steel Balancing block wool steel wool block and flask. Balancing block (b) (b) The steel wool is bned in (b) The a closed steel wool flask. is The bned block still in a closed balances. flask. The block The still balances. steel wool is bned in a closed flask. The block still balances. (c) (c) When the steel wool is bned (c) When in the open the steel flask, wool the is mass bned in in the the open flask flask, inceases. the mass When in the flask the steel inceases. wool is bned in the open flask, the mass in the flask inceases. We can se Newton s second law 1 a = F>m2 to elate the acceleation of a system object to the foces being exeted on it. Howeve, to se this law effectively we need qantitative infomation abot the foces that objects exet on each othe. Unfotnately, if two cas collide, we don t know the foce that one ca exets on the othe ding the collision. When fiewoks explode, we don t know the foces that ae exeted on the pieces flying apat. In this chapte yo will lean a new appoach that helps s analyze and pedict mechanical phenomena when the foces ae not known. 5.1 Mass acconting We begin o investigation by analyzing the physical qantity of mass. Ealie (in Chapte 2), we fond that the acceleation of an object depended on its mass the geate its mass, the less it acceleated de to an nbalanced extenal foce. We ignoed the possibility that an object s mass might change ding some pocess. Is the mass in a system always a constant vale? Yo have pobably obseved contless physical pocesses in which mass seems to change. Fo example, the mass of a log in a campfie deceases as the log bns; the mass of a seedling inceases as the plant gows. What happens to the lost mass fom the log? Whee does the seedling s inceased mass come fom? A system pespective helps s ndestand what happens to the bning log. If we choose only the log as the system, the mass of the system deceases as it bns. Howeve, ai is needed fo bning. What happens to the mass if we choose the sonding ai and the log as the system? Sppose that we place steel wool in a closed flask on one side of a balance scale and a metal block of eqal mass on the othe side (Fige 5.1a). In one expeiment, we bn the steel wool in the closed flask (the flask also contains ai), foming an oxide of ion. We find that the total mass of the closed flask containing bned steel wool (ion oxide) is the same as the mass of the balancing metal block (Fige 5.1b). Next, we bn the steel wool in an open flask and obseve that the mass of that flask inceases (Fige 5.1c). The steel wool in the open flask bns moe completely and absobs some extenal oxygen fom the ai as it bns. Eighteenth-centy Fench chemist Antoine Lavoisie actally pefomed sch expeiments. He ealized that the choice of the system was vey impotant. Lavoisie defined an isolated system as a gop of objects that inteact with each othe bt not with extenal objects otside the system. The mass of an isolated system is the sm of the masses of all objects in the system. He then sed the concept of an isolated system to smmaize his (and o) expeiments in the following way: Law of constancy of mass When a system of objects is isolated (a closed containe), its mass eqals the sm of the masses of its components and does not change it emains constant in time.

5.2 Linea momentm 153 When the system is not isolated (an open containe system), the mass might change. Howeve, this change is not andom it is always eqal to the amont of mass leaving o enteing the system fom the envionment. Ths, even when the mass of a system is not constant, we can keep tack of the changes if we take into accont how mch is leaving o enteing the system: initial mass of new mass enteing o final mass of system at ealie + leaving system between = system at late clock eading the two clock eadings clock eading The above eqation helps descibe the change of mass in any system. The mass is constant if thee is no flow of mass in o ot of the system, o the mass changes in a pedictable way if thee is some flow of mass between the system and the envionment. Basically, mass cannot appea fom nowhee and does not disappea withot a tace. Imagine yo have a system that has a total mass of m i = 3 kg (a bag of oanges). Yo add some moe oanges to the bag 1m = 1 kg2. The final mass of the system eqals exactly the sm of the initial mass and the added mass: m i + m = m f o 3 kg + 1 kg = 4 kg (Fige 5.2a). We can epesent this pocess with a ba chat (Fige 5.2b). The ba on the left epesents the initial mass of the system, the cental ba epesents the mass added o taken away, and the ba on the ight epesents the mass of the system in the final sitation. As a eslt, the height of the left ba pls the height of the cental ba eqals the height of the ight ba. The ba chat allows s to keep tack of the changes in mass of a system even if the system is not isolated. Mass is called a conseved qantity. A conseved qantity is constant in an isolated system. When the system is not isolated, we can accont fo the changes in the conseved qantity by what is added to o sbtacted fom the system. Jst as with evey idea in physics, the law of constancy of mass in an isolated system does not apply in all cases. We will discove late in this book (Chaptes 28 and 29) that in sitations involving atomic paticles, mass is not constant even in an isolated system; instead, what is constant is a new qantity that incldes mass as a component. Review Qestion 5.1 When yo bn a log in a fie pit, the mass of wood clealy deceases. How can yo define the system so as to have the mass of the objects in that system constant? 5.2 Linea momentm We now know that mass is an example of a conseved qantity. Is thee a qantity elated to motion that is conseved? When yo kick a stationay ball, thee seems to be a tansfe of motion fom yo foot to the ball. When yo knock bowling pins down with a bowling ball, a simila tansfe occs. Howeve, motion is not a physical qantity. What physical qantities descibing motion ae constant in an isolated system? Can we descibe the changes in these qantities sing a ba chat? Let s condct a few expeiments to find ot. In Obsevational Expeiment Table 5.1 we obseve two cats of diffeent masses that collide on a smooth tack. Fo these expeiments, the system will inclde both cats. A collision is a pocess that occs when two (o moe) objects come into diect contact with each othe. The system is isolated since the foces that the cats exet on each othe ae intenal, and extenal foces ae eithe balanced (as the vetical foces ae) o negligible (the hoizontal fiction foce). Fige 5.2 (a) The initial mass of the oanges pls the mass of the oanges that wee added (o sbtacted) eqals the final mass of the oanges. (b) The mass change pocess is epesented by a mass ba chat. (a) (b) m 3 kg 1 kg 4 kg m i m m f Active Leaning Gide

154 Chapte 5 Implse and Linea Momentm Obsevational Expeiment Table 5.1 Collisions in a system of two cats (all velocities ae with espect to the tack). Obsevational expeiment Analysis Video 5.1 Expeiment 1. Cat A (.2 kg) moving ight at 1. m/s collides with cat B (.2 kg), which is stationay. Cat A stops and cat B moves ight at 1. m/s. v Aix 1. m/s v Bix v Afx v Bfx 1. m/s A.2 kg B.2 kg A B x The diection of motion is indicated with a pls and a mins sign. Speed: The sm of the speeds of the system objects is the same befoe and afte the collision: 1. m>s + m>s = m>s + 1. m>s. Mass # speed: The sm of the podcts of mass and speed is the same befoe and afte the collision:.2 kg11. m>s2 +.2 kg1 m>s2 =.2 kg1 m>s2 +.2 kg11. m>s2. Mass # velocity: The sm of the podcts of mass and the x-component of velocity is the same befoe and afte the collision:.2 kg1+1. m>s2 +.2 kg12 =.2 kg12 +.2 kg1+1. m>s2. Expeiment 2. Cat A (.4 kg) moving ight at 1. m/s collides with cat B (.2 kg), which is stationay. Afte the collision, both cats move ight, cat B at 1.2 m/s, and cat A at.4 m/s. v Aix 1. m/s A.4 kg v Bix B.2 kg v Afx.4 m/s A v Bfx 1.2 m/s B x Speed: The sm of the speeds of the system objects is not the same befoe and afte the collision: 1. m>s + m>s.4 m>s + 1.2 m>s. Mass # speed: The sm of the podcts of mass and speed is the same befoe and afte the collision:.4 kg11. m>s2 +.2 kg1 m>s2 =.4 kg1.4 m>s2 +.2 kg11.2 m>s2. Mass # velocity: The sm of the podcts of mass and the x-component of velocity is the same befoe and afte the collision:.4 kg1+1. m>s2 +.2 kg12 =.4 kg1+.4 m>s2 +.2 kg1+1.2 m>s2. Expeiment 3. Cat A (.2 kg) with a piece of clay attached to the font moves ight at 1. m/s. Cat B (.2 kg) moves left at 1. m/s. The cats collide, stick togethe, and stop. v Aix 1. m/s v Bix 1. m/s v Afx v Bfx A.2 kg B.2 kg A B Speed: The sm of the speeds of the system objects is not the same befoe and afte the collision: 1. m>s + 1. m>s m>s + m>s. Mass # speed: The sm of the podcts of mass and speed is not the same befoe and afte the collision:.2 kg11. m>s2 +.2 kg11. m>s2.2 kg1 m>s2 +.2 kg1 m>s2. Mass # velocity: The sm of the podcts of mass and the x-component of velocity is the same befoe and afte the collision:.2 kg1+1. m>s2 +.2 kg1-1. m>s2 =.2 kg1 m>s2 +.2 kg1 m>s2. x Pattens One qantity emains the same befoe and afte the collision in each expeiment the sm of the podcts of the mass and x-velocity component of the system objects.

5.2 Linea momentm 155 In the thee expeiments in Obsevational Expeiment Table 5.1, only one qantity the sm of the podcts of mass and the x-component of velocity mv x emained the same befoe and afte the cats collided. Note also that the sm of the podcts of the mass and the y-component of velocity mv y did not change it emained zeo. Pehaps m v is the qantity chaacteizing motion that is constant in an isolated system. Bt will this patten pesist in othe sitations? Let s test this idea by sing it to pedict the otcome of the expeiment in Testing Expeiment Table 5.2. Active Leaning Gide Testing Expeiment table 5.2 Testing the idea that mv in an isolated system emains constant (all velocities ae with espect to the tack). Video 5.2 Testing expeiment Pediction Otcome Cat a (.4 kg) has a piece of modeling clay attached to its font and is moving ight at 1. m/s. Cat B (.2 kg) is moving left at 1. m/s. The cats collide and stick togethe. Pedict the velocity of the cats afte the collision. v Aix 1. m/s v fx? v Bix 1. m/s A.4 kg A B.2 kg B The system consists of the two cats. The diection of velocity is noted with a pls o mins sign of the velocity component: 1.4 kg21+1. m>s2 + 1.2 kg21-1. m>s2 x Afte the collision, the cats move togethe towad the ight at close to the pedicted speed. o = 1.4 kg +.2 kg2v f x v f x = 1+.2 kg # m>s2>1.6 kg2 = +.33 m>s Afte the collision, the two cats shold move ight at a speed of abot.33 m/s. Conclsion O pediction matched the otcome. This eslt gives s inceased confidence that this new qantity m v might be the qantity whose sm is constant in an isolated system. This new qantity is called linea momentm p. Linea Momentm The linea momentm p of a single object is the podct of its mass m and velocity v : p mv (5.1) Fige 5.3 Momentm is a vecto qantity with components. y m Linea momentm is a vecto qantity that points in the same diection as the object s velocity v (Fige 5.3). The SI nit of linea momentm is (kg # m/s). The total linea momentm of a system containing mltiple objects is the vecto sm of the momenta (plal of momentm) of the individal objects. p net m 1 v 1 m 2 v 2 P m n v n mv v p mv The components of a skydive s momentm: p x p y mv x

156 Chapte 5 Implse and Linea Momentm Note the following thee impotant points. 1. Unlike mass, which is a scala qantity, p = mv is a vecto qantity. Theefoe, it is impotant to conside the diection in which the colliding objects ae moving befoe and afte the collision. Fo example, becase cat B in Table 5.2 was moving left along the x-axis, the x-component of its momentm was negative befoe the collision. 2. Becase momentm depends on the velocity of the object, and the velocity depends on the choice of the efeence fame, diffeent obseves will mease diffeent momenta fo the same object. As a passenge, the momentm of a ca with espect to yo is zeo. Howeve, it is not zeo fo an obseve on the gond watching the ca move away fom him. 3. We chose an isolated system (the two cats) fo o investigation. The sm of the podcts of mass and velocity mv of all objects in the isolated system emained constant even thogh the cats collided with each othe. Howeve, if we had chosen the system to be jst one of the cats, we wold see that the linea momentm p = mv of the cat befoe the collision is diffeent than it is afte the collision. Ths, to establish that momentm p is a conseved qantity, we need to make se that the momentm of a system changes in a pedictable way fo systems that ae not isolated. We chose a system in Obsevational Expeiment Table 5.1 so that the sm of the extenal foces was zeo, making it an isolated system. Based on the eslts of Table 5.1 and Table 5.2, it appeas that the total momentm of an isolated system is constant. Momentm constancy of an isolated system The momentm of an isolated system is constant. Fo an isolated two-object system: m 1 v 1i m 2 v 2i m 1 v 1f m 2 v 2f (5.2) Becase momentm is a vecto qantity and Eq. (5.2) is a vecto eqation, we will wok with its x- and y-component foms: m 1 v 1i x m 2 v 2i x m 1 v 1f x m 2 v 2f x (5.3x) m 1 v 1i y m 2 v 2i y m 1 v 1f y m 2 v 2f y (5.3y) Fo a system with moe than two objects, we simply inclde a tem on each side of the eqation fo each object in the system. Let s test the idea that the momentm of an isolated system is constant in anothe sitation. Example 5.1 Two olleblades Jen (5 kg) and David (75 kg), both on olleblades, psh off each othe abptly. Each peson coasts backwad at appoximately constant speed. Ding a cetain time inteval, Jen tavels 3. m. How fa does David tavel ding that same time inteval? Sketch and tanslate The pocess is sketched at the ight. All motion is with espect to the floo and is along the x-axis. We choose the two olleblades as the system. Initially, the two olleblades ae at est. Afte pshing off, Jen (J) moves to the left and David (D) moves to the ight. We can se momentm constancy to calclate David s velocity component and pedict the distance he will tavel ding that same time inteval. Late Jen has taveled 3. m. Rolleblades psh off each othe. How fa has David taveled? Simplify and diagam We model each peson as a point-like object and assme that the fiction foce exeted on the olleblades does not affect thei motion. Ths thee ae no hoizontal extenal foces exeted on

5.3 Implse and momentm 157 the system. In addition, the two vetical foces, an pwad nomal foce N F on P that the floo exets on each peson and an eqalmagnitde downwad gavitational foce F E on P that Eath exets on each peson, cancel, as we see in the foce diagams. Since the net extenal foce exeted on the system is zeo, the system is isolated. The foces that the olleblades exet on each othe ae intenal foces and shold not affect the momentm of the system. Repesent mathematically The initial state (i) of the system is befoe they stat pshing on each othe, and the final state (f) is when Jen has taveled 3. m. m J v Ji x + m D v Di x = m J v Jf x + m D v Df x We choose the positive diection towad the ight. Becase the initial velocity of each peson is zeo, the above eqation becomes o + = m J v Jf x + m D v Df x Solve and evalate The x-component of Jen s velocity afte the psh-off is v Jf x = -13. m2> t, whee t is the time inteval needed fo he to tavel 3. m. We solve the above eqation fo David s final x-velocity component to detemine how fa David shold tavel ding that same time inteval: v Df x = - m Jv Jf x = - m J v m D m Jf x D 15 kg2 1-3. m2 = - = 175 kg2 t 12. m2 t Since momentm is constant in this isolated system, we pedict that David will tavel 2. m in the positive diection ding t. The meased vale is vey close to the pedicted vale. Ty it yoself: Estimate the magnitde of yo momentm when walking and when jogging. Assme yo mass is 6 kg. Answe: When walking, yo tavel at a speed of abot 1 to 2 m/s. So the magnitde of yo momentm will be p = mv 16 kg211.5 m>s2 9 kg # m>s. When jogging, yo speed is abot 2 to 5 m/s o a momentm of magnitde p = mv 16 kg213.5 m>s2 2 kg # m>s. m D v Df x = -m J v Jf x Notice that in Example 5.1 we wee able to detemine David s velocity by sing the pinciple of momentm constancy. We did not need any infomation abot the foces involved. This is a vey powefl eslt, since in all likelihood the foces they exeted on each othe wee not constant. The kinematics eqations we have sed p to this point have assmed constant acceleation of the system (and ths constant foces). Using the idea of momentm constancy has allowed s to analyze a sitation involving nonconstant foces. So fa, we have investigated sitations involving isolated systems. In the next section, we will investigate momentm in nonisolated systems. Review Qestion 5.2 Two identical cats ae taveling towad each othe at the same speed. One of the cats has a piece of modeling clay on its font. The cats collide, stick togethe, and stop. The momentm of each cat is now zeo. If the system incldes both cats, did the momentm of the system disappea? Explain yo answe. 5.3 Implse and momentm So fa, we have fond that the linea momentm of a system is constant if that system is isolated (the net extenal foce exeted on the system is zeo). How do we accont fo the change in momentm of a system when the net extenal foce exeted on it is not zeo? We can se Newton s laws to deive an expession elating foces and momentm change. Active Leaning Gide

158 Chapte 5 Implse and Linea Momentm Implse de to a foce exeted on a single object When yo psh a bowling ball, yo exet a foce on it, casing the ball to acceleate. The aveage acceleation a is defined as the change in velocity v f - v i divided by the time inteval t = t f - t i ding which that change occs: a = v f - v i t f - t i We can also se Newton s second law to detemine an object s acceleation if we know its mass and the sm of the foces that othe objects exet on it: a = F m We now have two expessions fo an object s acceleation. Setting these two expessions fo acceleation eqal to each othe, we get v f - v i t f - t i = F m Now mltiply both sides by m1t f - t i 2 and get the following: m v f - m v i = p f - p i = F1t f - t i 2 (5.4) The left side of the above eqation is the change in momentm of the object. This change depends on the podct of the net extenal foce and the time inteval ding which the foces ae exeted on the object (the ight side of the eqation). Note these two impotant points: 1. Eqation (5.4) is jst Newton s second law witten in a diffeent fom one that involves the physical qantity momentm. 2. Both foce and time inteval affect momentm the longe the time inteval, the geate the momentm change. A small foce exeted fo a long time inteval can change the momentm of an object by the same amont as a lage foce exeted fo a shot time inteval. Fige 5.4 The implse of a foce is the aea nde the F@vess@t gaph line. F (N) F av The podct of the extenal foce exeted on an object ding a time inteval and the time inteval gives s a new qantity, the implse of the foce. When yo kick a football o hit a baseball with a bat, yo foot o the bat exets an implse on the ball. The foces in these sitations ae not constant bt instead vay in time (see the example in Fige 5.4). The shaded aea nde the vaying foce cve epesents the implse of the foce. We can estimate the implse by dawing a hoizontal line that is appoximately the aveage foce exeted ding the time inteval of the implse. The aea nde the ectangla aveage foce-implse cve eqals the podct of the height of the ectangle (the aveage foce) and the width of the ectangle (the time inteval ove which the aveage foce is exeted). The podct F av 1t f - t i 2 eqals the magnitde of the implse. Implse The implse J of a foce is the podct of the aveage foce F av exeted on an object ding a time inteval 1t f - t i 2 and that time inteval: J Fav 1t f t i 2 (5.5) t i The aea of the shaded ectangle is abot the same as the aea nde the cved line and eqals the implse of the foce. t f t (s) Implse is a vecto qantity that points in the diection of the foce. The implse has a pls o mins sign depending on the oientation of the foce elative to a coodinate axis. The SI nit fo implse is N # s 1kg # m>s 2 2 # s kg # m>s, the same nit as momentm. It is often difficlt to mease diectly the implse of the net aveage foce ding a time inteval. Howeve, we can detemine the net foce on the ight

5.3 Implse and momentm 159 side of Eq. (5.4) indiectly by measing o calclating the momentm change on the left side of the eqation. Fo this eason, the combination of implse and momentm change povides a powefl tool fo analyzing inteactions between objects. We can now wite Eq. (5.4) as the implse-momentm eqation fo a single object. Implse-momentm eqation fo a single object If seveal extenal objects exet foces on a single-object system ding a time inteval 1t f t i 2, the sm of thei implses J cases a change in momentm of the system object: p f p i J F on System 1t f t i 2 (5.6) The x- and y-scala component foms of the implse-momentm eqation ae p f x p i x F on System x 1t f t i 2 (5.7x) p f y p i y F on System y 1t f t i 2 (5.7y) A few points ae woth emphasizing. Fist, notice that Eq. (5.6) is a vecto eqation, as both the momentm and the implse ae vecto qantities. Vecto eqations ae not easy to maniplate mathematically. Theefoe, we will se the scala component foms of Eq. (5.6) Eqs. (5.7x) and (5.7y). Second, the time inteval in the implse-momentm eqation is vey impotant. When object 2 exets a foce on object 1, the momentm of object 1 changes by an amont eqal to p 1 = p 1f - p 1i = F 2 on 1 1t f - t i 2 = F 2 on 1 t The longe that object 2 exets the foce on object 1, the geate the momentm change of object 1. This explains why a fast-moving object might have less of an effect on a stationay object ding a collision than a slow-moving object inteacting with the stationay object ove a longe time inteval. Fo example, a fast-moving bllet passing thogh a patially closed wooden doo might not open the doo (it will jst make a hole in the doo), wheeas yo little finge, moving mch slowe than the bllet, cold open the doo. Althogh the bllet moves at high speed and exets a lage foce on the doo, the time inteval ding which it inteacts with the doo is vey small (milliseconds). Hence, it exets a elatively small implse on the doo too small to significantly change the doo s momentm. A photo of a bllet shot thogh an apple illstates the effect of a shot implse time (Fige 5.5). The implse exeted by the bllet on the apple was too small to knock the apple off its sppot. Thid, if the magnitde of the foce changes ding the time inteval consideed in the pocess, we se the aveage foce. Finally, if the same amont of foce is exeted fo the same time inteval on a lage-mass object and on a small-mass object, the objects will have an eqal change in momentm (the same implse was exeted on them). Howeve, the small-mass object wold expeience a geate change in velocity than the lage-mass object. Fige 5.5 The bllet s time of inteaction with the apple is vey shot, casing a small implse that does not knock the apple ove. Active Leaning Gide Example 5.2 Abpt stop in a ca A 6-kg peson is taveling in a ca that is moving at 16 m/s with espect to the gond when the ca hits a baie. The peson is not weaing a seat belt, bt is stopped by an ai bag in a time inteval of.2 s. Detemine the aveage foce that the ai bag exets on the peson while stopping him. Sketch and tanslate Fist we daw an initial-final sketch of the pocess. We choose the peson as the system since we ae investigating a foce being exeted on him. (contined )

16 Chapte 5 Implse and Linea Momentm The peson s initial x-component of velocity v Pi x = +16 m>s deceases to the final x-component of velocity v Pf x = in a time inteval 1t f - t i 2 =.2 s. Ths the aveage foce exeted by the ai bag on the peson in the x-diection is 16 kg21-16 m>s2 F B on P x = 1.2 s - 2 = -48 N The negative sign in 48 N indicates that the aveage foce points in the negative x-diection. The magnitde of this foce is abot 1 lb! Simplify and diagam The foce diagam shows the aveage foce F B on P exeted in the negative diection by the bag on the peson. The vetical nomal foce and gavitational foces cancel. Repesent mathematically The x-component fom of the implse-momentm eqation is m P v Pi x + F B on P x 1t f - t i 2 = m P v Pf x Solve and evalate Solve fo the foce exeted by the ai bag on the peson: F B on P x = m P1v Pf x - v Pi x 2 1t f - t i 2 Ty it yoself: Sppose a 6-kg cash test dmmy is in a ca taveling at 16 m/s. The dmmy is not weaing a seat belt and the ca has no ai bags. Ding a collision, the dmmy flies fowad and stops when it hits the dashboad. The stopping time inteval fo the dmmy is.2 s. What is the aveage magnitde of the stopping foce that the dashboad exets on the dmmy? Answe: The aveage foce that the had sface exets on the dmmy wold be abot 5, N, extemely nsafe fo a hman. Note that the momentm change of the peson in Example 5.2 was the same. Howeve, since the change fo the dmmy occs ding a shote time inteval (.2 s instead of.2 s), the foce exeted on the dmmy is mch geate. This is why ai bags save lives. Fige 5.6 Analyzing the collision of two cats in ode to develop the momentm constancy idea. Initial Final v 1i Ding collision 1 2 F 2 on 1 v 1f F 1 on 2 1 2 v 2i v 2f 1 2 x x x Using Newton s laws to ndestand the constancy of momentm in an isolated system of two o moe objects Let s apply the implse-momentm eqation Eq. (5.4) to the scenaio we descibed in Obsevational Expeiment Table 5.1 in ode to exploe momentm constancy in a two-object isolated system. Two cats tavel towad each othe at diffeent speeds, collide, and ebond backwad (Fige 5.6). We fist analyze each cat as a sepaate system and then analyze them togethe as a single system. Assme that the vetical foces exeted on the cats ae balanced and that the fiction foce exeted by the sface on the cats does not significantly affect thei motion. Cat 1: In the initial state, befoe the collision, cat 1 with mass m 1 tavels in the positive diection at velocity v 1i. In the final state, afte the collision, cat 1 moves with a diffeent velocity v 1f in the opposite diection. To detemine the effect of the implse exeted by cat 2 on cat 1, we apply the implse-momentm eqation to cat 1 only: m 1 1 v 1f - v 1i 2 = F 2 on 1 1t f - t i 2

5.4 The genealized implse-momentm pinciple 161 Cat 2: We epeat this analysis with cat 2 as the system. Its velocity and momentm change becase of the implse exeted on it by cat 1: m 2 1 v 2f - v 2i 2 = F 1 on 2 1t f - t i 2 Newton s thid law povides a connection between o analyses of the two cats; inteacting objects at each instant exet eqal-magnitde bt oppositely diected foces on each othe: F 1 on 2 = -F 2 on 1 Sbstitting the expessions fo the foces fom above and simplifying, we get m 2 1v 2f - v 2i 2 = - m 11v 1f - v 1i 2 t f - t i t f - t i m 2 1v 2f - v 2i 2 = -m 1 1v 1f - v 1i 2 We now move the initial momentm fo both objects to the left side and the final momentm fo both objects to the ight side: m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f Initial momentm Final momentm This is the same eqation we aived at in Section 5.2, whee we obseved and analyzed collisions to ndestand the constant momentm of an isolated system. Hee we have eached the same conclsions sing only o knowledge of Newton s laws, momentm, and implse. Review Qestion 5.3 An apple is falling fom a tee. Why does its momentm change? Specify the extenal foce esponsible. Find a system in which the momentm is constant ding this pocess. 5.4 The genealized implse-momentm pinciple We can smmaize what we have leaned abot momentm in isolated and nonisolated systems. The change in momentm of a system is eqal to the net extenal implse exeted on it. If the net implse is zeo, then the momentm of the system is constant. This idea, expessed mathematically as the genealized implsemomentm pinciple, acconts fo sitations in which the system incldes one o moe objects and may o may not be isolated. The genealized implse-momentm pinciple means that we can teat momentm as a conseved qantity. Genealized implse-momentm pinciple Fo a system containing one o moe objects, the initial momentm of the system pls the sm of the implses that extenal objects exet on the system objects ding the time inteval 1t f - t i 2 eqals the final momentm of the system: 1m 1 v 1i m 2 v 2i P2 F on Sys 1t f t i 2 1m 1 v 1f m 2 v 2f P2 (5.8) Initial momentm of Net implse exeted on Final momentm of the system the system the system The x- and y-component foms of the genealized implse-momentm pinciple ae 1m 1 v 1i x m 2 v 2i x P2 F on Sys x 1t f t i 2 1m 1 v 1f x m 2 v 2f x P2 (5.9x) 1m 1 v 1i y m 2 v 2i y P2 F on Sys y 1t f t i 2 1m 1 v 1f y m 2 v 2f y P2 (5.9y) Note: If the net implse exeted in a paticla diection is zeo, then the component of the momentm of the system in that diection is constant.

162 Chapte 5 Implse and Linea Momentm Eqations (5.8) and (5.9) ae sefl in two ways. Fist, any time we choose to analyze a sitation sing the ideas of implse and momentm, we can stat fom a single pinciple, egadless of the sitation. Second, the eqations emind s that we need to conside all the inteactions between the envionment and the system that might case a change in the momentm of the system. Implse-momentm ba chats We can descibe an implse-momentm pocess mathematically sing Eqs. (5.9x and y). These eqations help s see that we can epesent the changes of a system s momentm sing a ba chat simila to the one sed to epesent the changes of a system s mass. The Reasoning Skill box shows the steps fo constcting an implse-momentm ba chat fo a simple system of two cats of eqal mass taveling towad each othe. Reasoning Skill Constcting a qalitative implse-momentm ba chat. 1. Sketch the pocess, choose the initial and final states, and choose a system. Initial Final v 1i v 2i v f 1 2 1 2 x p 1ix p 2ix J x p 1fx p 2fx 2. Daw initial and final momentm bas fo each object in the system. (Note cat diections and ba diections.) Slowe in negative diection (m 1 v 1ix m 2 v 2ix ) J x (m 1 v 1fx m 2 v 2fx ) o m 1 v 1i m 2 (v 2i ) (m 1 m 2 )v f 3. Daw an implse (J) ba if thee is an extenal nonzeo implse. 4. Convet each ba in the chat into a tem in the component fom of the implse-momentm eqation. Note that befoe constcting the ba chat, we epesent the pocess in an initial-final sketch (Step 1 in the Skill box). We then se the sketch to help constct the implse-momentm ba chat. The lengths of the bas ae qalitative indicatos of the elative magnitdes of the momenta. In the final state in the example shown, the cats ae stck togethe and ae moving in the positive diection. Since they have the same mass and velocity, they each have the same final momentm. The middle shaded colmn in the ba chat epesents the net extenal implse exeted on the system objects ding the time inteval 1t f - t i 2 thee is no implse fo the pocess shown. The shading eminds s that implse does not eside in the system; it is the inflence of the extenal objects on the momentm of the system. Notice that the sm of the heights of the bas on the left pls the height of the shaded implse ba shold eqal the sm of the heights of the bas on the ight. This consevation of ba heights eflects the consevation of momentm. We can se the ba chat to apply the genealized implse-momentm eqation (Step 4). Each nonzeo ba coesponds to a nonzeo tem in the eqation; the sign of the tem depends on the oientation of the ba.

5.4 The genealized implse-momentm pinciple 163 Using implse-momentm to investigate foces Can we se the ideas of implse and momentm to lean something abot the foces that two objects exet on each othe ding a collision? Conside a collision between two cas (Fige 5.7). To analyze the foce that each ca exets on the othe, we will define the system to inclde only one of the cas. Let s choose ca 1 and constct a ba chat fo it. Ca 2 exets an implse on ca 1 ding the collision that changes the momentm of ca 1. If the initial momentm of ca 1 is in the positive diection, then the implse exeted by ca 2 on ca 1 points in the negative diection. Becase of this, the implse ba on the ba chat points downwad. Note that the total height of the initial momentm ba on the left side of the chat and the height of the implse ba add p to the total height of the final momentm ba on the ight side. Using the ba chat, we can apply the component fom of the implse-momentm eqation: m 1 v 1i x + J x = m 1 v 1f x The components of the initial and final momentm ae positive. As the foce is exeted in the negative diection, the x-component of the implse is negative and eqal to -F 2 on 1 t. Ths, +m 1 v 1i + 1-F 2 on 1 t2 = +m 1 v 1f If we know the initial and final momentm of the ca and the time inteval of inteaction, we can se this eqation to detemine the magnitde of the aveage foce that ca 2 exeted on ca 1 ding the collision. Fige 5.7 A ba chat analysis of the collision of ca 2 with ca 1. v 1i Initial v 1f 1 2 1 2 1 2 m 1 m 1 Ca 1 has consideable momentm in the positive diection. F 2 on 1 p 1ix J 2 on 1x p 1fx The foce exeted by 2 on 1 is in the negative diection. Final Active Leaning Gide x Ca 1 has momentm in the positive diection. Tip When yo daw a ba chat, always specify the efeence fame (the object of efeence and the coodinate system). The diection of the bas on the ba chat (p fo positive and down fo negative) shold match the diection of the momentm o implse based on the chosen coodinate system. Example 5.3 Happy and sad balls Yo have two balls of identical mass and size that behave vey diffeently. When yo dop the so-called sad ball, it thds on the floo and does not bonce at all. When yo dop the so-called happy ball fom the same height, it bonces back to almost the same height fom which it was dopped. The diffeence in the boncing ability of the happy ball is de its intenal stcte; it is made of diffeent mateial. Yo hang each ball fom a sting of identical length and place a wood boad on its end diectly below the sppot fo each sting. Yo pll each ball back to an eqal height and elease the balls one at a time. When each ball hits the boad, which has the best chance of knocking the boad ove: the sad ball o the happy ball? Sad ball Happy ball Sketch and tanslate Initial and final sketches of the pocess ae shown at the ight. The system is jst the ball. In the initial state, the ball is jst abot to hit the boad, moving hoizontally towad the left (the balls ae moving eqally fast). The final state is jst afte the collision with the boad. The happy ball (H) bonces back, wheeas the sad ball (S) does not. Same initial Same momentm initial momentm fo both balls fo jst both befoe balls jst hitting befoe the hitting boad the boad Sad ball stops. Sad Happy ball ball stops. Happy bonces ball back. bonces back. (contined )

164 Chapte 5 Implse and Linea Momentm Simplify and diagam Assme that the collision time inteval t fo each ball is abot the same. We analyze only the hoizontal x-component of the pocess, the component that is elevant to whethe o not each of the boads is knocked ove. Each boad exets an implse on the ball that cases the momentm of the ball to change. Theefoe, each ball, accoding to Newton s thid law, exets an implse on the boad that it hits. A lage foce exeted on the boad means a lage implse and a bette chance to tip the boad. A ba chat fo each ball-boad collision is shown below. The implse of the happy ball is twice as lage in magnitde as that of the sad ball and cases twice as lage a momentm change. Repesent mathematically The x-component fom of the implse-momentm Eq. (5.5x) applied to each ball is as follows: Sad ball: mv i + F B on S x t = m # Happy ball: mv i + F B on H x t = m1-v i 2 Note that the x-component of the final velocity of the sad ball is v Sf x = (it does not bonce) and that the x-component of the final velocity of the happy ball is v Hf x = -v i (it bonces). Solve and evalate We can now get an expession fo the foce exeted by each boad on each ball: Sad ball: F B on S x = m1 - v i2 t Happy ball: F B on H x = m31-v i2 - v i 4 t = - mv i t. = - 2mv i t. Becase we assmed that the time of collision is the same, the boad exets twice the foce on the happy ball as on the sad ball, since the boad cases the happy ball s momentm to change by an amont twice that of the sad ball. Accoding to Newton s thid law, this means that the happy ball will exet twice as lage a foce on the boad as the sad ball. Ths, the happy ball has a geate chance of tipping the boad. Ty it yoself: Is it less safe fo a football playe to bonce backwad off a goal post o to hit the goal post and stop? Answe: Althogh any collision is dangeos, it is bette to hit the goal post and stop. If the football playe bonces back off the goal post, his momentm will have changed by a geate amont (like the happy ball in the last example). This means that the goal post exets a geate foce on him, which means thee is a geate chance fo injy. The patten we fond in the example above is te fo all collisions when an object bonces back afte a collision, we know that a lage magnitde foce is exeted on it than if the object had stopped and did not bonce afte the collision. Fo that eason, blletpoof vests fo law enfocement agents ae designed so that the bllet embeds in the vest athe than boncing off it. Review Qestion 5.4 If in solving the poblem in Example 5.3 we chose the system to be the ball and the boad, how wold the mathematical desciption fo each ball-boad collision change? 5.5 Skills fo analyzing poblems sing the implse-momentm eqation Initial and final sketches and ba chats ae sefl tools to help analyze pocesses sing the implse-momentm pinciple. Let s investigate fthe how these tools wok togethe. A geneal stategy fo analyzing sch pocesses is

5.5 Skills fo analyzing poblems sing the implse-momentm eqation 165 descibed on the left side of the table in Example 5.4 and illstated on the ight side fo a specific pocess. PROBLEM-SOLvinG STRATEGY Applying the implse-momentm eqation Active Leaning Gide Sketch and tanslate Sketch the initial and final states and inclde appopiate coodinate axes. Label the sketches with the known infomation. Decide on the object of efeence. Choose a system based on the qantity yo ae inteested in; fo example, a mlti-object isolated system to detemine the velocity of an object, o a singleobject nonisolated system to detemine an implse o foce. Example 5.4 Bllet hits wood block A.2-kg bllet taveling hoizontally at 25 m>s embeds in a 1.-kg block of wood esting on a table. Detemine the speed of the bllet and wood block togethe immediately afte the bllet embeds in the block. The left side of the sketch below shows the bllet taveling in the positive x-diection with espect to the gond; it then joins the wood. All motion is along the x-axis; the object of efeence is Eath. The system incldes the bllet and wood; it is an isolated system since the vetical foces balance. The initial state is immediately befoe the collision; the final state is immediately afte. Initial m B.2 kg, m W 1. kg v Bix 25 m/s, v Wix v Bi Final v B-Wfx? v B-Wf x x Simplify and diagam Detemine if thee ae any extenal implses exeted on the system. Dawing a foce diagam cold help detemine the extenal foces and thei diections. Daw an implse-momentm ba chat fo the system fo the chosen diection(s) to help yo ndestand the sitation, fomlate a mathematical epesentation of the pocess, and evalate yo eslts. Assme that the fiction foce exeted by the tabletop on the bottom of the wood does not change the momentm of the system ding the vey shot collision time inteval. The ba chat epesents the pocess. The ba fo the bllet is shote than that fo the block thei velocities ae the same afte the collision, bt the mass of the bllet is mch smalle. We do not daw a foce diagam hee, as the system is isolated. p Bix p Wix J x p Bfx p Wfx Repesent mathematically Use the ba chat to apply the genealized implse-momentm eqation along the chosen axis. Each nonzeo ba becomes a nonzeo tem in the eqation. The oientation of the ba detemines the sign in font of the coesponding tem in the eqation. Remembe that momentm and implse ae vecto qantities, so inclde the pls o mins signs of the components based on the chosen coodinate system. m B v Bi x + m W # + 1Jx 2 = m B v B@Wf x + m W v B@Wf x Since J x =, v B@Wf x = m B v Bi x 1m B + m W 2 (contined )

166 Chapte 5 Implse and Linea Momentm Solve and evalate Inset the known infomation to detemine the nknown qantity. Check if yo answe is easonable with espect to sign, nit, and magnitde. Also make se it applies fo limiting cases, sch as objects of vey small o vey lage mass. v B@Wf x = 1.2 kg2125 m>s2 1.2 kg + 1. kg2 = +4.9 m>s The magnitde of the answe seems easonable given how fast the bllet was initially taveling. The pls sign indicates the diection, which makes sense, too. The nits ae also coect (m/s). We can test this sing a limiting case: if the mass o speed of the bllet is zeo, the block emains stationay afte the collision. Ty it yoself: A.2-kg bllet is fied hoizontally into a 2.-kg block of wood esting on a table. Immediately afte the bllet joins the block, the block and bllet move in the positive x-diection at 4. m/s. What was the initial speed of the bllet? Answe: 4 m/s. We cold have woked Example 5.4 backwad to detemine the initial speed of the bllet befoe hitting the block (like the Ty It Yoself qestion). This execise wold be sefl since the bllet tavels so fast that it is difficlt to mease its speed. Vaiations of this method ae sed, fo example, to decide whethe o not golf balls confom to the necessay les. The balls ae hit by the same mechanical lanching implse and the moving balls embed in anothe object. The balls speeds ae detemined by measing the speed of the object they embed in. Detemining the stopping time inteval fom the stopping distance When a system object collides with anothe object and stops a ca collides with a tee o a wall, a peson jmps and lands on a solid sface, o a meteoite collides with Eath the system object tavels what is called its stopping distance. By estimating the stopping distance of the system object, we can estimate the stopping time inteval. Sppose that a ca ns into a lage tee and its font end cmples abot.5 m. This.5 m, the distance that the cente of the ca taveled fom the beginning of the impact to the end, is the ca s stopping distance. Similaly, the depth of the hole left by a meteoite povides a ogh estimate of its stopping distance when it collided with Eath. Howeve, to se the implsemomentm pinciple, we need the stopping time inteval associated with the collision, not the stopping distance. Hee s how we can se a known stopping distance to estimate the stopping time inteval. Assme that the acceleation of the object while stopping is constant. In that case, the aveage velocity of the object while stopping is jst the sm of the initial and final velocities divided by 2: v aveage x = 1v f x + v i x 2>2. Ths, the stopping displacement 1x f - x i 2 and the stopping time inteval 1t f - t i 2 ae elated by the kinematics eqation x f - x i = v aveage x 1t f - t i 2 = 1v f x + v i x 2 1t 2 f - t i 2

5.5 Skills fo analyzing poblems sing the implse-momentm eqation 167 Reaange this eqation to detemine the stopping time inteval: t f - t i = 21x f - x i 2 v f x + v i x (5.1) Eqation (5.1) povides a method to convet stopping distance x f - x i into stopping time inteval t f - t i. Eqation (5.1) can be applied to hoizontal o vetical stopping. Active Leaning Gide Example 5.5 Stopping the fall of a movie stnt dive The ecod fo the highest movie stnt fall withot a paachte is 71 m (23 ft), held by 8-kg A. J. Baknas. His fall was stopped by a lage ai cshion, into which he sank abot 4. m. His speed was abot 36 m/s (8 mi/h) when he eached the top of the ai cshion. Estimate the aveage foce that the cshion exeted on his body while stopping him. Sketch and tanslate We focs only on the pat of the fall when Baknas is sinking into the cshion. The sitation is sketched below. We choose Baknas as the system and the y-axis pointing p. The initial state is jst as he toches the cshion at position y i = +4. m, and the final state is when the cshion has stopped him, at position y f =. All motion is with espect to Eath. The othe infomation abot the pocess is given in the fige. Be se to pay attention to the signs of the qantities (especially the initial velocity). Each extenal foce cases an implse. Baknas has zeo momentm in the final state. Repesent mathematically Since all motion and all of the foces ae in the vetical diection, we se the ba chat to help constct the vetical y-component fom of the implse-momentm eqation [Eq. (5.6y)] to detemine the foce that the cshion exets on Baknas as he sinks into it: m B v i y + 1N C on B y + F E on B y 21t f - t i 2 = m B v f y Simplify and diagam We daw a foce diagam top ight, modeling Baknas as a point-like object. Since Baknas s downwad speed deceases, the cshion mst be exeting an pwad foce on Baknas of geate magnitde than the downwad foce that Eath exets on him. Ths, the net foce exeted on him points pwad, in the positive y- diection. Using this infomation, we can daw a qalitative implse-momentm ba chat fo the pocess. Using the foce diagam, we see that the y-components of the foces ae N C on B y = +N C on B and F E on B y = -F E on B = -m B g, whee N C on B is the magnitde of the aveage nomal foce that the cshion exets on Baknas, the foce we ae tying to estimate. Noting that v f y = and sbstitting the foce components into the above eqation, we get m B v i y + 31+N C on B 2 + 1-m B g241t f - t i 2 = m B # 1 m B v i y + 1N C on B - m B g21t f - t i 2 =. We can find the time inteval that the cshion takes to stop Baknas sing Eq. (5.1) and noting that v f y = : t f - t i = 21y f - y i 2 + v i y (contined )

168 Chapte 5 Implse and Linea Momentm Solve and evalate The stopping time inteval while Baknas sinks 4. m into the cshion is t f - t i = Solving fo N C on B, we get N C on B = -m Bv i y 1t f - t i 2 + m Bg = 21-4. m2 + 1-36 m>s2 =.22 s -18 kg21-36 m>s2 1.22 s2 + 18 kg219.8 N>kg2 = +13, N + 78 N = 14, N Wow, that is a hge foce! To edce the isk of injy, stnt dives pactice landing so that the stopping foce that a cshion exets on them is distibted evenly ove the entie body. The cshions mst be deep enogh so that they povide a long stopping time inteval and ths a smalle stopping foce. The same stategy is applied to developing ai bags and collapsible fames fo atomobiles to make them safe fo passenges ding collisions. Notice fo impotant points. Fist, we ve inclded only two significant digits since that is how many the data had. Second, it is vey easy to make sign mistakes. A good way to avoid these is to daw a sketch that incldes a coodinate system and labels showing the vales of known physical qantities, inclding thei signs. Thid, the implse de to Eath s gavitational foce is small in magnitde compaed to the implse exeted by the ai cshion. Lastly, the foce exeted by the ai cshion wold be even geate if the stopping distance and conseqently the stopping time inteval wee shote. Ty it yoself: Sppose that the cshion in the last example stopped Baknas in 1. m instead of 4. m. What wold be the stopping time inteval and the magnitde of the aveage foce of the cshion on Baknas? Answe: The stopping time inteval is.56 s, and the aveage stopping foce is appoximately 5, N. Ode-of-magnitde estimate will bone beak? The stategy that we sed in the pevios example can be sed to analyze skll facte injies that might lead to concssions. Laboatoy expeiments indicate that the hman skll can facte if the compessive foce exeted on it pe nit aea is 1.7 * 1 8 N>m 2. The sface aea of the skll is mch smalle than 1 m 2, so we will se sqae centimetes, a moe easonable nit of aea fo this discssion. Since 1 m 2 = 1 * 1 4 cm 2, we convet the compessive foce pe aea to 11.7 * 1 8 N>m 2 1 m 2 2a 1 * 1 4 cm 2 b = 1.7 * 14 N>cm 2. Example 5.6 Bone facte estimation 1 A bicyclist is watching fo taffic fom the left while tning towad the ight. A steet sign hit by an ealie ca accident is bent ove the side of the oad. The cyclist s head hits the pole holding the sign. Is thee a significant chance that his skll will facte? S k e t c h a n d tanslate The p o c e s s i s sketched at the ight. The initial state is at the instant that the h e a d i n i t i a l l y contacts the pole; the final state is when the head and body have stopped. The peson is the system. We have been given little infomation, so we ll have to make some easonable estimates of vaios qantities in ode to make a decision abot a possible skll facte. Simplify and diagam The ba chat illstates the momentm change of the system and the implse exeted by the pole that cased the change. The peson was initially moving in the hoizontal x- diection with espect to Eath, and not moving afte the collision. The pole exeted an implse in the negative x- diection on the cyclist. We ll need to estimate the following qantities: the mass and speed of the cyclist in this sitation, the stopping time inteval, and the aea of contact. Let s assme that this is a 7-kg cyclist moving at abot 3 m/s. The peson s body keeps moving fowad fo a shot distance afte the bone makes contact with the pole. The skin indents some ding the collision. Becase of these two factos, we assme 1 This is a te stoy it happened to one of the book s athos, Alan Van Hevelen.

5.6 Jet poplsion 169 a stopping distance of abot 1 cm. Finally, we assme an aea of contact of abot 4 cm 2. All of these nmbes have lage ncetainties and we ae not woying abot significant figes, becase this is jst an estimate. The pole s negative implse on the peson cases the momentm to decease. Repesent mathematically We now apply the genealized implse-momentm pinciple: m Peson v Peson i x + 1F Pole on Peson x 21t f - t i 2 = m Peson v Peson f x = 1 F Pole on Peson x = - m Pesonv Peson i x t f - t i We can se the stategy fom the last example to estimate the stopping time inteval t f - t i fom the stopping distance x f - x i : t f - t i = 21x f - x i 2 v f x + v i x whee v ix is the initial velocity of the cyclist and v fx = is his final velocity. Solve and evalate Sbstitting the estimated initial velocity and the stopping distance into the above, we get an estimate fo the stopping time inteval: t f - t i = 21x f - x i 2 v f x + v i x = 21.1 m - 2 =.67 s + 3 m>s Since this stopping time inteval is an intemediate calclated vale, we don t need to woy abot its nmbe of significant digits. When we complete o estimate, thogh, we will keep jst one significant digit. We can now inset o estimated vales of qantities in the expession fo the foce exeted by the pole on the peson: F Pole on Peson x = - m Pesonv Peson i x 1t f - t i 2 17 kg213 m>s2 = - = -3 N 1.67 s2 O estimate of the foce pe aea is Foce Aea = 3 N 4 cm 2 8 N>cm 2 Is the peson likely to facte his skll? The foce pe aea needed to beak a bone is abot 1.7 * 1 4 N>cm 2 = 17, N>cm 2. O estimate cold have been off by at least a facto of 1. The foce pe aea is still too little fo a facte. Ty it yoself: What wold be the magnitde of the foce exeted on the cyclist if he bonced back off the pole instead of stopping, assming the collision time inteval emains the same? Answe: 6 N. Review Qestion 5.5 As the bllet entes the block in Example 5.4, the block exets a foce on the bllet, casing the bllet s speed to decease to almost zeo. Why did we not inclde the implse exeted by the block on the bllet in o analysis of this sitation? 5.6 Jet poplsion Cas change velocity becase of an inteaction between the ties and the oad. Likewise, a ship s popelles psh wate backwad; in tn, wate pshes the ship fowad. Once the ship o ca is moving, the extenal foce exeted by the wate o the oad has to balance the opposing fiction foce o the vehicle s velocity will change. What does a ocket psh against in empty space to change its velocity? Rockets cay fel that they ignite and then eject at high speed ot of the exhast nozzles (see Fige 5.8). Cold this bning fel ejected fom the ocket povide the psh to change its velocity? Choose the system to be the ocket and fel togethe. If the ocket and fel ae at est befoe the ocket fies its engines, then its momentm is zeo. If thee ae no extenal implses, then even afte the ocket fies its engines, the momentm of the ocket-fel system shold still be zeo. Howeve, the bning fel is ejected backwad at high velocity fom the exhast nozzle and has a backwad momentm. The ocket mst now have a nonzeo fowad velocity. We test this idea qantitatively in Testing Expeiment Table 5.3. Fige 5.8 A ocket as it expels fel. Expelled fel moves left. Rocket moves ight.

17 Chapte 5 Implse and Linea Momentm Testing Expeiment table 5.3 Rocket poplsion. Testing expeiment Pediction Otcome Yo ae taveling thogh space in a ocket and obseve anothe ocket moving with eqal velocity next to yo. All of a sdden yo notice a bst of bning fel that is ejected fom it. Pedict what happens to that ocket s velocity. Choose the othe ocket and its fel as the system. Yo ocket seves as the object of efeence and the +x-diection is in the diection of its motion. The othe ocket has zeo velocity in the initial state with espect to the object of efeence. Its final state is jst afte it expels fel backwad at high speed; the ocket in tn gains an eqal magnitde of momentm in the fowad diection. We can epesent this pocess with an initial-final sketch and a momentm ba chat fo the ocket-fel system. Initial v Initial m Fel m Rocket x The velocity of the othe ocket does incease, and we see it move ahead of o ocket. Final m Fel m Rocket v Fel v Rocket Fel and ocket system p Rix p Fix J x p Rfx p Ffx Assming that fel is ejected all at once at constant speed, the velocity of the ocket shold be = m Rocket v Rocket x + m Fel v Fel x 1 v Rocket x = - m Felv Fel x m Rocket Hee m Rocket is the mass of the ocket withot the fel. We can also choose the ocket alone as the system. The ocket pshes back on the fel, expelling it backwad at high speed 1v Fel x 6 2; the fel in tn pshes fowad against the ocket, exeting an implse that cases the ocket s momentm and the velocity (assming the mass of the ocket itself does not change) to incease 1v Rocket x 7 2. Rocket alone as system p Rix J F on Rx p Rfx Conclsion The otcome of the expeiment is consistent with the pediction, sppoting the genealized implse-momentm pinciple. We have leaned that, independent of the choice of the system, when a ocket expels fel in one diection, it gains velocity and theefoe momentm in the opposite diection. This mechanism of acceleating a ocket o spaceship is called jet poplsion.

5.7 Meteoites, adioactive decay, and two-dimensional collisions: Ptting it all togethe 171 The foce exeted by the fel on a ocket ding jet poplsion is called thst. Typical lage ocket thsts mease in mega-newtons 11 6 N2, and exhast speeds ae moe than 1 times the speed of sond. Thst povides the necessay implse to change a ocket s momentm. Yo can obseve the pinciples of jet poplsion sing a long, naow balloon. Blow p the balloon; then open the valve and elease it. The balloon will shoot away apidly in the opposite diection of the ai steaming ot of the balloon s valve. In eality, a ocket bns its fel gadally athe than in one shot bst; ths its mass is not a constant nmbe bt changes gadally. Howeve, the same methods we sed in Testing Expeiment Table 5.3, togethe with some calcls, can be applied to detemine the change in the ocket s velocity. The main idea behind the jet poplsion method is that when an object ejects some of its mass in one diection, it acceleates in the opposite diection. This means that the same method that is sed to speed p a ocket can also be sed to slow it down. To do this, the fel needs to be ejected in the same diection that the ocket is taveling. Tip Yo can become yo own jet poplsion machine by standing on olleblades o a skateboad and thowing a medicine ball o a heavy book fowad o backwad. Review Qestion 5.6 The following eqation is a soltion fo a poblem. State a possible poblem. 12. kg21-8. m>s2 + = 12. kg + 58 kg2v x. 5.7 Meteoites, adioactive decay, and two-dimensional collisions: Ptting it all togethe In this section we apply implse-momentm ideas to analyze meteoites colliding with Eath, adioactive decay of adon in the lngs, and two-dimensional ca collisions. We stat by analyzing a eal meteoite collision with Eath that occed abot 5, yeas ago. Canyon Diablo Cate In this example we se two sepaate choices of systems to answe diffeent qestions abot a meteoite collision with Eath. Example 5.7 Meteoite impact Aizona s Meteo Cate (also called Canyon Diablo Cate), shown in Fige 5.9, was podced 5, yeas ago by the impact of a 3 * 1 8 -kg meteoite taveling at 1.3 * 1 4 m>s (29, mi/h). The cate is abot 2 m deep. Estimate (a) the change in Eath s velocity as a eslt of the impact and (b) the aveage foce exeted by the meteoite on Eath ding the collision. Sketch and tanslate A sketch of the pocess is shown on the next page. To analyze Eath s motion, we choose a coodinate system at est with espect to Eath. The oigin of the coodinate axis is at the point whee the meteoite fist hits Eath. We keep tack of the dot at the bottom of the meteoite. The axis points in the diection of the meteoite s motion. Fige 5.9 Canyon Diablo Cate, site of a meteoite impact 5, yeas ago. To answe the fist qestion, we choose Eath and the meteoite as the system and se momentm constancy to detemine Eath s change in velocity de to the (contined )

172 Chapte 5 Implse and Linea Momentm constancy to detemine the speeds of Eath and the meteoite afte they join togethe: 1m E # + mm v Mi y 2 + 31t f - t i 24 = 1m E v Ef y + m M v Mf y 2 Keep tack of this point. 1 m M v Mi y = m E v E f y + m M v Mf y = 1m E + m M 2v f y To estimate the foce that the meteoite exets on Eath ding the collision, we se the y-component fom of the implse-momentm eqation with the meteoite alone as the system: m M v Mi y + F E on M y 1t f - t i 2 = m M v Mf y The time inteval eqied fo the collision [sing Eq. (5.9)] is collision. To estimate the aveage foce that the meteoite exeted on Eath ding the collision (and that Eath exeted on the meteoite), we choose the meteoite alone as the system and se the implse-momentm eqation to answe that qestion. Simplify and diagam Assme that the meteoite hits pependicla to Eath s sface in the positive y-diection. The fist implse-momentm ba chat below epesents the pocess fo the Eath-meteoite system to answe the fist qestion. The second ba chat epesents the meteoite alone as the system ding its collision with Eath to answe the second qestion. Isolated system: momentm is constant. Eath s implse on the meteoite cases the meteoite s momentm to decease. Repesent mathematically The y-component of the meteoite s initial velocity is v Mi y = +1.3 * 1 4 m>s. Eath s initial velocity is zeo (with espect to the object of efeence). The y-component of the meteoite s final velocity eqals Eath s since the meteoite embeds in Eath. The meteoite s mass is abot 3 * 1 8 kg and Eath s mass is 6 * 1 24 kg. We se momentm t f - t i = 21y f - y i 2 v Mf y + v Mi y Solve and evalate To answe the fist qestion, we solve fo the final velocity of Eath and meteoite togethe: v fy = = m M m E + m M v Mi y 3 * 1 8 kg 6 * 1 24 kg + 3 * 1 8 kg 11.3 * 14 m>s2 = 7 * 1-13 m>s This is so slow that it wold take Eath abot 5, yeas to tavel jst 1 m. Since Eath is so mch moe massive than the meteoite, the meteoite s impact has extemely little effect on Eath s motion. Fo the second qestion, the time inteval fo the impact is abot t f - t i = 21y f - y i 2 v Mf y + v Mi y = 212 m2 11.3 * 1 4 m>s + 7 * 1-13 m>s2 =.31 s Like most implsive collisions, this one was ove qickly! Note that we ve estimated the displacement of the meteoite to be the depth of the cate. Reaanging the implse-momentm eqation as applied to the collision, we find the aveage foce exeted by Eath on the meteoite: F E on M y = m M1v Mf y - v Mi y 2 1t f - t i 2 = 13 * 18 kg217 * 1-13 m>s - 1.3 * 1 4 m>s2 1.31 s2 = -1.3 * 1 14 N -1 * 1 14 N The foce exeted by Eath on the meteoite is negative it points opposite the diection of the meteoite s initial

5.7 Meteoites, adioactive decay, and two-dimensional collisions: Ptting it all togethe 173 velocity. Accoding to Newton s thid law, the foce that the meteoite exets on Eath is positive and has the same magnitde: F M on E y = +1 * 1 14 N This sonds like a vey lage foce, bt since the mass of Eath is 6 * 1 24 kg, this foce will case an acceleation of a little ove 1-11 m>s 2, a vey small nmbe. Ty it yoself: Estimate the change in Eath s velocity and acceleation if it wee hit by a meteoite taveling at the same speed as in the last example, stopping in the same distance, bt having mass of 6 * 1 19 kg instead of 3 * 1 8 kg. Answe: Abot.1 m/s and 4 m>s 2. Tip Notice how the choice of system in Example 5.7 was motivated by the qestion being investigated. Always think abot yo goal when deciding what yo system will be. An object beaks into pats (adioactive decay) We will lean in the chapte on nclea physics (Chapte 28) that the nclei of some atoms ae nstable and spontaneosly beak apat. In a pocess called alpha decay, the ncles of the atom beaks into a daghte ncles that is slightly smalle and lighte than the oiginal paent ncles and an even smalle alpha paticle (symbolized by a; actally a helim ncles). Fo example, adon decays into a polonim ncles (the daghte) and an alpha paticle. Radon is podced by a seies of decay eactions stating with heavy elements in the soil, sch as anim. Radon diffses ot of the soil and can ente a home thogh cacks in its fondation, whee it can be inhaled by people living thee. Once in the lngs, the adon ndegoes alpha decay, eleasing fastmoving alpha paticles that may case mtations that cold lead to cance. In the next example, we will analyze alpha decay by adon by sing the idea of momentm constancy. Example 5.8 Radioactive decay of adon in lngs An inhaled adioactive adon ncles esides moe o less at est in a peson s lngs, whee it decays to a polonim ncles and an alpha paticle. With what speed does the alpha paticle move if the polonim ncles moves away at 4. * 1 5 m>s elative to the lng tisse? The mass of the polonim ncles is 54 times geate than the mass of the alpha paticle. Sketch and tanslate An initial-final sketch of the sitation is shown at the ight. We choose the system to be the adon ncles in the initial state, which convets to the polonim ncles and the alpha paticle in the final state. The coodinate system has a positive x-axis pointing in the diection of motion of the alpha paticle, with the object of efeence being the lng tisse. The initial velocity of the adon ncles along the x-axis is and the final velocity component of the polonim daghte ncles is v Po fx = -4. * 1 5 m>s. The final velocity component of the alpha paticle v af x is nknown. If the mass of the alpha paticle is m, then the mass of the polonim is 54m. (contined )

174 Chapte 5 Implse and Linea Momentm Simplify and diagam Assme that thee ae no extenal foces exeted on the system, meaning that the system is isolated and ths its momentm is constant. The implsemomentm ba chat below epesents the pocess. Repesent mathematically Use the ba chat to help apply the implse-momentm eqation fo the pocess: m Rn 12 + 121t f - t i 2 = m Po v Pof x + m a v af x 1 = m Po v Pof x + m a v af x Solve and evalate Reaanging, we get an expession fo the final velocity of the alpha paticle in the x-diection: v af x = - m Pov Pof x m a The x-component of the velocity of the alpha paticle afte adon decay is v af x = - m Pov Pof x m a = +2.2 * 1 7 m>s Isolated system: momentm is constant. = - 154m a21-4. * 1 5 m>s2 m a The sign indicates that the alpha paticle is taveling in the positive x-diection opposite the diection of the polonim. The magnitde of this velocity is hge abot one-tenth the speed of light! The speeding alpha paticle passes thogh lng tisse and collides with atoms and molecles, dislodging electons and ceating ions. Radon expose cases appoximately 15, cases of lng cance each yea. Ty it yoself: Fancim nclei ndego adioactive decay by emitting eithe an alpha paticle o a beta paticle (an electon). The alpha paticle is abot 8 times moe massive than a beta paticle. If the paticles ae emitted with the same speed, in which case is the ecoil speed of the ncles that is left afte an alpha o a beta paticle is emitted geatest? Answe: Since the mass of the alpha paticle is mch geate than the mass of the beta paticle, and they ae taveling with the same speed, the momentm of the alpha paticle is mch geate than the momentm of the beta paticle. Theefoe, the ncles that is left wold have a geate ecoil speed ding alpha decay. Collisions in two dimensions So fa, the collisions we have investigated have occed along one axis. Often, a moto vehicle accident involves two vehicles taveling along pependicla paths. Fo these two-dimensional collisions, we can still apply the ideas of implse and momentm, bt we will se one implse-momentm eqation fo each coodinate axis. Example 5.9 A 16-kg pickp tck taveling east at 2 m/s collides with a 13-kg ca taveling noth at 16 m/s. The vehicles emain tangled togethe afte the collision. Detemine the velocity (magnitde and diection) of the combined weck immediately afte the collision. Sketch and tanslate We sketch the initial and final sitations of the vehicles. We se a P sbscipt fo the pickp and a C sbscipt fo the ca. The initial state is jst befoe the collision; the final state is jst afte the vehicles collide and ae moving togethe. We choose the two vehicles as the system. The object of efeence is Eath; the positive x-axis points east and the positive y-axis points noth. Simplify and diagam Foce diagams epesent the side view fo each vehicle jst befoe the collision. We

5.7 Meteoites, adioactive decay, and two-dimensional collisions: Ptting it all togethe 175 assme that the fiction foce exeted by the oad is vey small compaed to the foce that each vehicle exets on the othe. Ths, we ignoe the implse de to sface fiction ding the shot collision time inteval of abot.1 s. We then apply momentm constancy in each diection. Implse-momentm ba chats fo the x-diection and fo the y-diection ae shown below. The net extenal foce on the system is zeo an isolated system. The x and y-components of momentm ae independently constant. y-component eqation: 116 kg21 m>s2 + 113 kg2116 m>s2 = 129 kg2v P@Cf sin Divide the left side of the second eqation by the left side of the fist eqation and the ight side of the second eqation by the ight of the fist, and cancel the 29 kg and v P + Cf on the top and bottom of the ight side. We get 113 kg2116 m>s2 116 kg212 m>s2 = sin cos = tan =.65 A 33 angle has a.65 tangent. Ths, the vehicles move off at 33 above the +x-axis (the noth of east diection). We can now se this angle with eithe the x-component eqation o the y-component eqation above to detemine the speed of the two vehicles immediately afte the collision. Using the x-component eqation, we get 116 kg212 m>s2 v P@Cf = = 13 m>s 129 kg2cos 33 Fom the y-component eqation, we have v P@Cf = 113 kg2116 m>s2 129 kg2sin 33 = 13 m>s Repesent mathematically Now, convet each momentm ba in the x-component ba chat into a tem in the x-component fom of the implse-momentm eqation [Eq. (5.7x)] and each ba in the y-component ba chat into a tem in the y-component fom of the implsemomentm eqation [Eq. (5.7y)]. Notice that the x-component of the final velocity vecto is v P@Cf x = v P@Cf cos and the y-component is v P + Cf x = v P@Cf sin : x@component eqation: m P v Pi x + m C v Ci x = 1m P + m C 2v P@Cf cos y@component eqation: m P v Pi y + m C v Ci y = 1m P + m C 2v P@Cf sin We have two eqations and two nknowns 1v P@Cf and 2. We can solve fo both nknowns. Solve and evalate x-component eqation: 116 kg212 m>s2 + 113 kg21 m>s2 = 129 kg2v P@Cf cos The two eqations give the same eslt fo the final speed, a good consistency check. Fo collisions in which vehicles lock togethe like this, police investigatos commonly se the lengths of the skid maks along with the diection of the vehicles afte the collision to detemine thei initial speeds. This allows them to decide whethe eithe vehicle was exceeding the speed limit befoe the collision. Ty it yoself: Use a limiting case analysis and the x- and y-component foms of the implse-momentm eqation to pedict what wold happen ding the collision if the pickp had infinite mass. Is the answe easonable? Answe: If we place in 113 kg2116 m>s2 116 kg212 m>s2 = sin cos = tan in place of the 16-kg mass of the pickp, the left side of the eqation becomes zeo. Then tan =. The pickp wold move staight ahead when hitting the ca. In othe wods, the collision with the ca wold not change the diection of tavel of the pickp. The eslt seems easonable if the mass of the pickp was lage compaed to the mass of the ca. Review Qestion 5.7 When a meteoite hits Eath, the meteoite s motion appaently disappeas completely. How can we claim that momentm is conseved?

Smmay Wods Isolated system An isolated system is one in which the objects inteact only with each othe and not with the envionment, o the sm of extenal foces exeted on it is zeo. (Sections 5.1 5.2) Isolated system Nonisolated system Pictoial and physical epesentations Mathematical epesentation Consevation of mass If the system is isolated, its mass is constant. If the system is not isolated, the change in the system s mass eqals the mass deliveed to the system o taken away fom it. (Section 5.1) m 3 kg 1 kg 4 kg m i + m = m f m i m m f Linea momentm p is a vecto qantity that is the podct of an object s mass m and velocity v. The total momentm of the system is the sm of the momenta of all objects in the system. (Section 5.2) Implse J Implse is the podct of the aveage extenal foce F av exeted on an object ding a time inteval t and that time inteval. (Section 5.3) p v m p = mv Eq. (5.1) p system = p 1 + p 2 + g J = Fav 1t f - t i 2 Eq. (5.5) 176

Qestions 177 Genealized implse-momentm pinciple If the system is isolated, its momentm is constant. If the system is not isolated, the change in the system s momentm eqals the sm of the implses exeted on the system ding the time inteval t = 1t f - t i 2. (Section 5.4) v 1i Initial v 2i 1 2 v 1i p 1ix p 2ix J x p 1fx p 2fx Final v 1f v 2f v 1f 1 2 1m 1 v 1i + m 2 v 2i + g2 + F on Sys t = 1m 1 v 1f + m 2 v 2f + g2 Eq. (5.8) x- and y-component foms: m 1 v 1i x + m 2 v 2i x + F on Sys x t = m 1 v 1f x + m 2 v 2f x Eq. (5.9x) m 1 v 1i y + m 2 v 2i y + F on Sys y t = m 1 v 1f y + m 2 v 2f y Eq. (5.9y) 1 2 1 2 p 1ix J 2 on 1x p 1fx Fo instcto-assigned homewok, go to www.masteingphysics.com Qestions Mltiple Choice Qestions 1. The gavitational foce that Eath exets on an object cases an implse of +1 N # s in one expeiment and +1 N # s on the same object in anothe expeiment. How can this be? (a) The mass of the obje ct changed. (b) The time intevals ding which the foce was exeted ae diffeent. (c) The magnitdes of the foce wee diffeent. 2. A bllet fied at a doo makes a hole in the doo bt does not open it. Yo finge does not make a hole in the doo bt does open it. Why? (a) The bllet is too small. (b) The foce exeted by the bllet is not enogh to open the doo. (c) A finge exets a smalle foce bt the time inteval is mch longe. (d) The bllet goes thogh the doo and does not exet a foce at all. 3. How wold yo convince somebody that the momentm of an isolated system is constant? (a) It is a law; ths, yo do not need to convince anybody. (b) Use an example fom a textbook to show that the sm of the initial and final velocities of the objects involved in a collision ae the same. (c) Deive it fom Newton s second and thid laws. (d) Use it to make pedictions abot a new expeiment, and then compae the otcome to the pediction. (e) Both (c) and (d) will wok. 4. A wagon fll of medicine balls is olling along a steet. Sddenly one medicine ball (3 kg) falls off the wagon. What happens to the speed of the wagon? (a) The wagon slows down. (b) The speed of the wagon does not change. (c) The wagon speeds p. (d) Additional infomation abot the ball s motion is needed to answe. 5. When can yo apply the idea that momentm is constant to solve a poblem? (a) When the system is isolated (b) When the system is not isolated bt the time inteval when the extenal foces ae exeted is vey small (c) When the extenal foces ae mch smalle than the intenal foces 6. Choose an example in which the momentm of a system is not constant. (a) A bllet shot fom a ifle, with the ifle and the bllet as the system (b) A feely falling metal ball, with the ball as the system (c) A feely falling metal ball, with the ball and Eath as the system (d) It is not possible to give an example since the momentm of a system is always constant.

178 Chapte 5 Implse and Linea Momentm 7. Why do cannons oll back afte each shot? (a) A cannon pshes on a shell and the shell pshes back on the cannon. (b) The momentm of the cannon-shell system is constant. (c) Both a and b ae coect. 8. Which is a safe ca bmpe in a collision: one that is flexible and etacts o one that is igid? Why? (a) The etactable bmpe, becase softe things withstand collisions bette (b) The etactable bmpe, becase it extends the time inteval of the collision, ths edcing the foce exeted on the ca (c) The igid bmpe, becase it does not change shape so easily 9. Why does an inflated balloon shoot acoss a oom when ai is eleased fom it? (a) Becase the otside ai pshes on the balloon (b) Becase the momentm of the balloon-ai system is constant (c) Becase the ai inside the balloon pshes on the balloon, exeting the same foce that the balloon exets on the ai (d) Both b and c ae coect. 1. In which sitation does the momentm of a tennis ball change moe? (a) It hits the acket and stops. (b) It hits the acket and flies off in the opposite diection. (c) It misses the acket and contines moving. 11. A toy ca with vey low fiction wheels and axles ests on a level tack. In which sitation will its speed incease moe? (a) It is hit fom the ea by a wad of clay that sticks to the ca. (b) It is hit by a bbe ball with the same mass and velocity of the clay that ebonds in the opposite diection afte hitting the ca. 12. A meteoite stikes Eath and foms a cate, deceasing the meteoite s momentm to zeo. Does this phenomenon contadict the consevation of momentm? Choose as many answes as yo think ae coect. (a) No, becase the meteoite system is not isolated (b) No, becase in the meteoite-eath system, Eath acqies momentm lost by the meteoite (c) No, becase the meteoite bings momentm fom space (d) Yes, becase the meteoite is not moving elative to a medim befoe the collision 13. A 1-kg ca taveling east at 24 m/s collides with a 2-kg ca taveling west at 21 m/s. The cas lock togethe. What is thei velocity immediately afte the collision? (a) 3 m/s east (b) 3 m/s west (c) 6 m/s east (d) 6 m/s west (e) 15 m/s east Conceptal Qestions 14. Accoding to a epot on tamatic bain injy, woodpeckes smack thei heads against tees at a foce eqivalent to 12 g s withot sffeing bain damage. This statement contains one o moe mistakes. Identify the mistakes in this statement. 15. Jim says that momentm is not a conseved qantity becase objects can gain and lose momentm. Do yo agee o disagee? If yo disagee, what can yo do to convince Jim of yo opinion? 16. Say five impotant things abot momentm (fo example, momentm is a vecto qantity). How does each statement apply to eal life? 17. Thee people ae obseving the same ca. One peson claims that the ca s momentm is positive, anothe peson claims that it is negative, and the thid peson says that it is zeo. Can they all be ight at the same time? Explain. 18. When wold a ball hitting a wall have a geate change in momentm: when it hits the wall and bonces back at the same speed o when it hits and sticks to the wall? Explain yo answe. 19. In the pevios qestion, in which case does the wall exet a geate foce on the ball? Explain. 2. Explain the diffeence between the concepts of constancy and consevation. Povide an example of a conseved qantity and a nonconseved qantity. 21. Why do yo believe that momentm is a conseved qantity? 22. A heavy ba falls staight down onto the bed of a olling tck. What happens to the momentm of the tck at the instant the ba lands on it? Explain. How many coect answes do yo think ae possible? Make se yo think of what falls staight down means. 23. Constct implse-momentm ba chats to epesent a falling ball in (a) a system whose momentm is not constant and (b) a system whose momentm is constant. In the initial state, the ball is at est; in the final state, the ball is moving. 24. A peson moving on olleblades thows a medicine ball in the diection opposite to he motion. Constct an implsemomentm ba chat fo this pocess. The peson is the system. 25. A peson moving on olleblades dops a medicine ball staight down elative to himself. Constct an implsemomentm ba chat fo the system consisting of the ball and Eath fo this pocess. The olleblade is the object of efeence, and the final state is jst befoe the ball hits the gond. Poblems Below, BIO indicates a poblem with a biological o medical focs. Poblems labeled EST ask yo to estimate the answe to a qantitative poblem athe than deive a specific answe. Asteisks indicate the level of difficlty of the poblem. Poblems with no * ae consideed to be the least difficlt. a single * maks intemediate difficlt poblems. Two ** indicate moe difficlt poblems. 5.2 Linea momentm 1. Yo and a fiend ae playing tennis. (a) What is the magnitde of the momentm of the.57-kg tennis ball when it tavels at a speed of 3 m/s? (b) At what speed mst yo.32-kg tennis acket move to have the same magnitde momentm as the ball? (c) If yo n towad the ball at a speed of 5. m/s, and the ball is flying diectly at yo at a speed of 3 m/s, what

Poblems 179 is the magnitde of the total momentm of the system (yo and the ball)? Assme yo mass is 6 kg. In evey case specify the object of efeence. 2. Yo ae hitting a tennis ball against a wall. The.57-kg tennis ball taveling at 25 m/s stikes the wall and ebonds at the same speed. (a) Detemine the ball s oiginal momentm (magnitde and diection). (b) Detemine the ball s change in momentm (magnitde and diection). What is yo object of efeence? 3. A ball of mass m and speed v tavels hoizontally, hits a wall, and ebonds. Anothe ball of the same mass and taveling at the same speed hits the wall and sticks to it. Which ball has a geate change in momentm as a eslt of the collision? Explain yo answe. 4. (a) A 145-g baseball tavels at 35 m/s towad a baseball playe s bat (the bat is the object of efeence) and ebonds in the opposite diection at 4 m/s. Detemine the ball s momentm change (magnitde and diection). (b) A golfe hits a.46-kg golf ball that lanches fom the gass at a speed of 5 m/s. Detemine the ball s change in momentm. 5. * A 13-kg ca is taveling at a speed of 1 m/s with espect to the gond when the dive acceleates to make a geen light. The momentm of the ca inceases by 12,8 kg # m>s. List all the qantities yo can detemine sing this infomation and detemine thee of those qantities. 6. * The les of tennis specify that the.57-kg ball mst bonce to a height of between 53 and 58 inches when dopped fom a height of 1 inches onto a concete slab. What is the change in the momentm of the ball ding the collision with the concete? Yo will have to se some fee-fall kinematics to help answe this qestion. 7. A cat of mass m moving ight at speed v with espect to the tack collides with a cat of mass.7m moving left. What is the initial speed of the second cat if afte the collision the cats stick togethe and stop? 8. A cat of mass m moving ight collides with an identical cat moving ight at half the speed. The cats stick togethe. What is thei speed afte the collision? 9. EST Estimate yo momentm when yo ae walking at yo nomal pace. 5.3 Implse and momentm 1. A 1-g apple is falling fom a tee. What is the implse that Eath exets on it ding the fist.5 s of its fall? The next.5 s? 11. * The same 1-g apple is falling fom the tee. What is the implse that Eath exets on it ding the fist.5 m of its fall? The next.5 m? 12. Why does Eath exet the same implse ding the two time intevals in Poblem 1 bt diffeent implses ding the same distances taveled in Poblem 11? 13. * Van hits concete sppot In a cash test, a van collides with a concete sppot. The stopping time inteval fo the collision is.1 s, and the implse exeted by the sppot on the van is 7.5 * 1 3 N # s. (a) Detemine eveything yo can abot the collision sing this infomation. (b) If the van is constcted to collapse moe ding the collision so that the time inteval ding which the implse is exeted is tipled, what is the aveage foce exeted by the concete sppot on the van? 14. BIO Foce exeted by heat on blood Abot 8 g of blood is pmped fom a peson s heat into the aota ding each heatbeat. The blood stats at est with espect to the body and has a speed of abot 1. m/s in the aota. If the pmping takes.17 s, what is the magnitde of the aveage foce exeted by the heat on the blood? 15. * The tain tacks on which a tain tavels exet a 2. * 1 5 N fiction foce on the tain, casing it to stop in 5 s. (a) Detemine the aveage foce needed to stop the tain in 25 s. (b) Detemine the stopping time inteval if the tacks exet a 1. * 1 5 @N fiction foce on the tain. 16. ** EST Yo fiend is catching a falling basketball afte it has passed thogh the basket. He hands move staight down.2 m while catching the ball. Estimate (a) the time inteval fo the ball to stop as she catches it and (b) the aveage foce that he hands exet on the ball while catching it. Indicate any assmptions o estimates yo have to make in ode to answe the qestions. 17. * BIO Tamatic bain injy Accoding to a epot on tamatic bain injy, the foce that a pofessional boxe s fist exets on his opponent s head is eqivalent to being hit with a 5.9 kg bowling ball taveling at 8.9 m/s that stops in.18 s. Detemine the aveage foce that the fist exets on the head. 18. * A 65-kg astonat pshes against the inside back wall of a 2-kg spaceship and moves towad the font. He speed inceases fom to 1.6 m/s. (a) If he psh lasts.3 s, what is the aveage foce that the astonat exets on the wall of the spaceship? (b) If the spaceship was initially at est, with what speed does it ecoil? (c) What was the object of efeence that yo sed to answe pats (a) and (b)? 19. * Yo decide to se yo gaden hose to wash yo gaage doo. The wate shoots ot at a ate of 1 kg/s and a speed of 16 m/s with espect to the hose. When the wate hits the gaage, its speed deceases to zeo. Detemine the foce that the wate exets on the wall. What assmptions did yo make? 2. * The ai in a windstom moves at a speed of 3 m/s. When it hits a stop sign, the ai stops momentaily. The mass of ai hitting the stop sign each second is abot 2. kg. Make a list of physical qantities yo can detemine sing this infomation and detemine thee of them. 21. * An egg olls off a kitchen conte and beaks as it hits the floo. How lage is the implse that the floo exets on the egg, and how lage is the foce exeted on the egg by the floo when stopping it? The conte is 1. m high, the mass of the egg is abot 5 g, and the time inteval ding the collision is abot.1 s. 22. ** Retactable ca bmpe A ca bmpe exets an aveage foce on a ca as it etacts a cetain distance ding a collision. Using the implse-momentm eqation, show that the magnitde of the foce and the etaction distance ae elated by the eqation Fx =.5mv 2. What assmptions did yo make? 23. ** Popotional easoning Use popotional easoning and the eqation fom Poblem 22 to detemine (a) the necessay pecent change in the etaction distance so that the aveage foce eqied to stop a ca is edced by 2% and (b) the pecent change in initial to final speed that wold podce the same edction in foce. 24. (a) What foce is eqied to stop a 15-kg ca in a distance of.2 m if it is initially moving at 2.2 m/s? (b) What if the ca is moving at 4.5 m/s?

18 Chapte 5 Implse and Linea Momentm 25. * A boxe delives a pnch to his opponent s head, which has a mass of 7. kg. Use the gaph in Fige P5. 25 to estimate (a) the implse of the foce exeted by the boxe and (b) the speed of the head afte the pnch is deliveed. What assmptions did yo make? 26. * Ai bag foce on head The gaph in Fige P5.26 shows the time vaiation of the foce that an atomobile s ai bag exets on a peson s head ding a collision. The mass of the head is 8. kg. Detemine (a) the total implse of the foce exeted by the ai Fige P5.25 Foce (N) 5 3 1.5.1 Time (s) Fige P5.26 Ai bag Foce (N) 2 1.15 bag on the peson s head and (b) the peson s speed jst befoe the collision occed. 27. * Eqation Jeopady 1 Invent a poblem fo which the soltion is 127 kg21-3. m>s2 + 13 kg21+4. m>s2 = 127 kg + 3 kg2v. 28. * Eqation Jeopady 2 Invent a poblem fo which the soltion is 1.2 kg213 m>s2-11 N21.4 s2 = 1.2 kg211 m>s2. 29. * Wite a geneal implse-momentm eqation that descibes the following pocess: a peson skating on olleblades eleases a backpack that falls towad the gond (the pocess ends befoe the backpack hits the gond). What is the system, and what ae the physical qantities yo will se to descibe the pocess? 5.4 Genealized implse-momentm pinciple 3. * Two cats (1 g and 15 g) on an ai tack ae sepaated by a compessed sping. The sping is eleased. Repesent the pocess with a momentm ba chat (a) with one cat as the system and (b) with both cats as the system. (c) Wite expessions fo all of the physical qantities yo can fom this infomation. Identify yo object of efeence. 31. * A tennis ball of mass m hits a wall at speed v and ebonds at abot the same speed. Repesent the pocess with an implse-momentm ba chat fo the ball as the system. Using the ba chat, develop an expession fo the change in the ball s momentm. What is the object of efeence? 32. * A tennis ball taveling at a speed of v stops afte hitting a net. Repesent the pocess with an implse-momentm ba chat fo the ball as the system. Develop an expession fo the ball s change in momentm. What is the object of efeence? 33. * Yo dop a happy ball and a sad ball of the same mass fom height h (see Fige 5.1). One ball hits the gond and ebonds almost to the oiginal height. The othe ball does not 5 1 Time (ms) bonce. Repesent each pocess with a ba chat, stating jst befoe the balls hit the gond to jst afte the fist ball ebonds and when the othe ball stops. Choose the ball as the system. 34. * Yo expeiment again with the balls fom Poblem 33. Yo dop them fom the same height onto a le that is placed on the edge of a table (Fige P5.34). One ball knocks the le off; the othe does not. Repesent each pocess with an implse-momentm ba chat with (a) the ball as a system and (b) the ball and the le as the system. The pocess stats jst befoe the balls hit the le and ends immediately afte they hit the le. Use the ba chats to help explain the diffeence in the eslts of the expeiment. 35. ** Yo demonstate hitting a boad in a kaate class. The speed of yo hand as it hits the thick boad is 14 m/s with espect to the boad, and the mass of yo hand is abot.8 kg. How deep does yo hand go into the boad befoe stopping if the collision lasts fo 2. * 1-3 s? What assmptions did yo make? What othe qantities can yo detemine sing this infomation? 36. * Yo hold a beach ball with yo ams extended above yo head and then thow it pwad. Repesent the motion of the ball with an implse-momentm ba chat fo (a) the ball as the system and (b) the ball and Eath as the system. 37. * A basketball playe dops a.6-kg basketball vetically so that it is taveling at 6. m/s when it eaches the floo. The ball ebonds pwad at a speed of 4.2 m/s. (a) Detemine the magnitde and diection of the ball s change in momentm. (b) Detemine the aveage net foce that the floo exets on the ball if the collision lasts.12 s. 38. * Ba chat Jeopady Invent a poblem fo each of the ba chats shown in Fige P5.38. Fige P5.34 Fige P5.38 5.5 Skills fo solving implse-momentm poblems 39. * A baseball bat contacts a.145-kg baseball fo 1.3 * 1-3 s. The aveage foce exeted by the bat on the ball is 89 N. If the ball has an initial velocity of 36 m/s towad the bat and the foce of the bat cases the ball s motion to evese diection, what is the ball s speed as it leaves the bat? (a) (b) p Oix J F on Ox p Ofx v i p 1ix p 2ix J x p 1fx p 2fx

Poblems 181 4. * A tennis ball taveling hoizontally at a speed of 4. m/s hits a wall and ebonds in the opposite diection. The time inteval fo the collision is abot.13 s, and the mass of the ball is.59 kg. Make a list of qantities yo can detemine sing this infomation and detemine fo of them. Assme that the ball ebonds at the same speed. 41. A cannon monted on the back of a ship fies a 5-kg cannonball in the hoizontal diection at a speed of 15 m/s. If the cannon and ship have a combined mass of 4 kg and ae initially at est, what is the speed of the ship jst afte shooting the cannon? What assmptions did yo make? 42. * A team in Qebec is playing ice baseball. A 72-kg playe who is initially at est catches a 145-g ball taveling at 18 m/s. If the playe s skates ae fictionless, how mch time is eqied fo him to glide 5. m afte catching the ball? 43. A 1-kg sled caying a 3-kg child glides on a hoizontal, fictionless sface at a speed of 6. m/s towad the east. The child jmps off the back of the sled, popelling it fowad at 2 m/s. What was the child s velocity in the hoizontal diection elative to the gond at the instant she left the sled? 44. A 1,-kg coal ca on the Geat Nothen Raiload coasts nde a coal stoage bin at a speed of 2. m/s. As it goes nde the bin, 1 kg of coal is dopped into the ca. What is the final speed of the loaded ca? 45. * Avoiding chest injy A peson in a ca ding a sdden stop can expeience potentially seios chest injies if the combined foce exeted by the seat belt and sholde stap exceeds 16, N. Descibe what it wold take to avoid injy by estimating (a) the minimm stopping time inteval and (b) the coesponding stopping distance, assming an initial speed of 16 m/s. Indicate any othe assmptions yo made. 46. * Bising apples An apple bises if a foce geate than 8. N is exeted on it. Wold a.1-kg apple be likely to bise if it falls 2. m and stops afte sinking.6 m into the gass? Explain. 47. * Fast tennis seve The fastest seve in women s tennis is Vens Williams, who ecoded a seve of 24 km/h at the Fench Open in 27. Sppose that the mass of he acket was 328 g and the mass of the ball was 57 g. If he acket was moving at 2 km/h when it hit the ball, appoximately what was the acket s speed afte hitting the ball? Indicate any assmptions yo made. 48. * Yo ae in an elevato whose cable has jst boken. The elevato is falling at 2 m/s when it stats to hit a shockabsobing device at the bottom of the elevato shaft. If yo ae to avoid injy, the pwad foce that the floo of the elevato exets on yo pight body while stopping shold be no moe than 8 N. Detemine the minimm stopping distance needed to avoid injy (do not foget to inclde yo mass in the calclations). What assmptions did yo make? Do these assmptions make the stopping distance smalle o lage than the eal-wold vale? 49. * Yo jmp fom the window of a bning hotel and land in a safety net that stops yo fall in 1. m. Estimate the aveage foce that the net exets on yo if yo ente the net at a speed of 24 m/s. What assmptions did yo make? If yo did not make these assmptions, wold the stopping distance be smalle o lage? 5. * Skid maks A ca skids to a stop. The length of the skid maks is 5 m. What infomation do yo need in ode to decide whethe the ca was speeding befoe the dive hit the bakes? 51. * BIO Leg injies ding ca collisions Ding a ca collision, the knee, thighbone, and hip can sstain a foce no geate than 4 N. Foces that exceed this amont cold case dislocations o factes. Assme that in a collision a knee stops when it hits the ca s dashboad. Also assme that the mass of the body pats stopped by the knee is abot 2% of the total body mass. (a) What minimm stopping time inteval in needed to avoid injy to the knee if the peson is initially taveling at 15 m/s (34 mi/h)? (b) What is the minimm stopping distance? 52. * BIO Bone facte The zygomatic bone in the ppe pat of the cheek can be facted by a 9-N foce lasting 6. ms o longe. A hockey pck can easily exet sch a foce when hitting an npotected face. (a) What change in velocity of a.17-kg hockey pck is needed to povide that implsive foce? What assmptions did yo make? (b) A padded facemask dobles the stopping time. By how mch does it change the foce on the face? Explain. 53. * An implse of 15 N # s stops yo head ding a ca collision. (a) A cash test dmmy s head stops in.2 s, when the cheekbone hits the steeing wheel. What is the aveage foce that the wheel exets on the dmmy s cheekbone? (b) Wold this cash facte a hman cheekbone (see Poblem 52)? (c) What is the shotest impact time that a peson cold sstain withot beaking the bone? 54. A cat is moving on a hoizontal tack when a heavy bag falls vetically onto it. What happens to the speed of the cat? Repesent the pocess with an implse-momentm ba chat. 55. * A cat is moving on a hoizontal tack. A heavy bag falls off the cat and moves staight down elative to the cat. Descibe what happens to the speed of the cat. Repesent yo answe with the implse-momentm ba chat. [Hint: What efeence fame will yo se when yo daw the ba chat?] 5.6 and 5.7 Jet poplsion and Ptting it all togethe 56. Yo fiend shoots an 8-g aow thogh a 1-g apple balanced on William Tell s head. The aow has a speed of 5 m/s befoe passing thogh the apple and 4 m/s afte. Detemine the final speed of the apple. 57. * BIO Potassim decay in body tisse Cetain natal foms of potassim have nclei that ae adioactive. Each adioactive potassim ncles decays to a slightly less massive daghte ncles and a high-speed electon called a beta paticle. If afte the decay the daghte ncles is moving at speed 2 m/s with espect to the decaying mateial, how fast is the electon (the beta paticle) moving? Indicate any assmptions yo made. The mass of the daghte is abot 7, times geate than the mass of the beta paticle. 58. ** Meteoite impact with Eath Abot 65 million yeas ago a 1-km-diamete 2 * 1 15 -kg meteoite taveling at abot 1 km/s cashed into what is now the Glf of Mexico. The impact podced a clod of debis that dakened Eath and led to the extinction of the dinosas. Estimate the speed Eath gained as a eslt of this impact and the aveage foce that the meteoite exeted on Eath ding the collision. Indicate any assmptions made in yo calclations. 59. ** Thee fiends play beach volleyball. The 28 g ball is flying east at speed 8. m/s with espect to the gond when one of the playes bmps the ball noth. The foce exeted by the wist on the ball has an aveage magnitde of 84 N and lasts fo.1 s. Detemine the ball s velocity (magnitde and diection) following the bmp. Does yo answe make sense?

182 Chapte 5 Implse and Linea Momentm 6. * Ca collision A 118-kg ca taveling soth at 24 m/s with espect to the gond collides with and attaches to a 247-kg delivey tck taveling east at 16 m/s. Detemine the velocity (magnitde and diection) of the two vehicles when locked togethe jst afte the collision. 61. * Ice skates collide While ice skating, yo nintentionally cash into a peson. Yo mass is 6 kg, and yo ae taveling east at 8. m/s with espect to the ice. The mass of the othe peson is 8 kg, and he is taveling noth at 9. m/s with espect to the ice. Yo hang on to each othe afte the collision. In what diection and at what speed ae yo taveling jst afte the collision? 62. Difting space mechanic An astonat with a mass of 9 kg (inclding spacesit and eqipment) is difting away fom his spaceship at a speed of.2 m/s with espect to the spaceship. The astonat is eqipped only with a.5-kg wench to help him get back to the ship. With what speed and in what diection elative to the spaceship mst he thow the wench fo his body to acqie a speed of.1 m/s and diect him back towad the spaceship? Explain. 63. * Astonat flings oxygen tank While the astonat in Poblem 62 is tying to get back to the spaceship, his comade, a 6-kg astonat, is floating at est a distance of 1 m fom the spaceship when she ns ot of oxygen and fel to powe he back to the spaceship. She emoves he oxygen tank (3. kg) and flings it away fom the ship at a speed of 15 m/s elative to the ship. (a) At what speed elative to the ship does she ecoil towad the spaceship? (b) How long mst she hold he beath befoe eaching the ship? 64. Rocket stages A 5-kg ocket ejects a 1,-kg package of fel. Befoe ejection, the ocket and the fel tavel togethe at a speed of 2 m/s with espect to distant stas. If afte the ejection, the fel package tavels at 5 m/s opposite the diection of its initial motion, what is the velocity of the ocket? 65. * A ocket has jst ejected fel. With the fel and the ocket as the system, constct an implse-momentm ba chat fo (a) the ocket s incease in speed and (b) the pocess of a ocket slowing down de to fel ejection. (c) Finally, daw ba chats fo both sitations sing the ocket withot the fel as the system. 66. ** Yo have two cats, a foce pobe connected to a compte, a motion detecto, and an assotment of objects of diffeent masses. Design thee expeiments to test whethe momentm is a conseved qantity. Descibe caeflly what data yo will collect and how yo will analyze the data. Geneal Poblems 67. ** EST Estimate the ecoil speed of Eath if all of the inhabitants of Canada and the United States simltaneosly jmped staight pwad fom Eath s sface (eaching heights fom seveal centimetes to a mete o moe). Indicate any assmptions that yo made in yo estimate. 68. * A cat of mass m taveling in the negative x-diection at speed v collides head-on with a cat that has tiple the mass and is moving at 6% of the speed of the fist cat. The cats stick togethe afte the collision. In which diection and at what speed will they move? 69. ** Two cas of neqal mass moving at the same speed collide head-on. Explain why a passenge in the smalle mass ca is moe likely to be injed than one in the lage mass ca. Jstify yo easoning with the help of physics pinciples. 7. * Restaining foce ding collision A 134-kg ca taveling east at 13.6 m/s (2 mi/h) has a head-on collision with a 193-kg ca taveling west at 2.5 m/s (3 mi/h). If the collision time is.1 s, what is the foce needed to estain a 68-kg peson in the smalle ca? In the lage ca? 71. ** E S T A capente hammes a nail sing a.4-kg hammehead. Pat of the nail goes into a boad. (a) Estimate the speed of the hammehead befoe it hits the nail. (b) Estimate the stopping distance of the hammehead. (c) Estimate the stopping time inteval. (d) Estimate the aveage foce that the hammehead exets on the nail. 72. ** A.2-kg bllet taveling at a speed of 3 m/s embeds in a 1.-kg wooden block esting on a hoizontal sface. The block slides hoizontally 4. m on a sface befoe stopping. Detemine the coefficient of kinetic fiction between the block and sface. 73. ** EST Nolan Ryan may be the fastest baseball pitche of all time. The Ginness Book of Wold Recods clocked his fastball at 1.9 mi/h in a 1974 game against the Chicago White Sox. Use the implse-momentm eqation to estimate the foce that Ryan exeted on the ball while thowing that pitch. Inclde any assmptions yo made. 74. ** A ecod ainstom podced 34.8 mm (appoximately 1 ft) of ain in 42 min. Estimate the aveage foce that the ain exeted on the oof of a hose that meases 1 m * 16 m. Indicate any assmptions yo made. 75. ** EST The U.S. Amy special nits MH-47E helicopte has a mass of 23, kg, and its popelle blades sweep ot an aea of 263 m 2. It is able to hove at a fixed elevation above one landing point by pshing ai downwad (the ai pshes p on the helicopte blades). Choose a easonable ai mass displaced downwad each second and the speed of that ai in ode fo the helicopte to hove. Indicate any assmptions yo made. 76. ** A 245-kg spots tility vehicle hits the ea end of a 122-kg ca at est at a stop sign. The ca and SUV emain locked togethe and skid 4.6 m befoe stopping. If the coefficient of fiction between the vehicles and the oad is.7, what was the SUV s initial velocity? 77. ** A ca of mass m 1 taveling noth at a speed of v 1 collides with a ca of mass m 2 taveling east at a speed of v 2. They lock togethe afte the collision. Develop expessions fo the diection and the distance the cas will move ntil they stop if the coefficient of kinetic fiction m k between the cas ties and the oad is abot the same fo both cas. 78. ** Foce exeted by wind on Willis Towe A 1.-m/s wind blows against one side of the Willis Towe in Chicago. The bilding is 443 m tall and appoximately 8 m wide. Estimate the aveage foce of the ai on the side of the bilding. The density of ai is appoximately 1.3 kg>m 3. Indicate any assmptions that yo made. 79. * Wite yo own poblem. Wite and solve a poblem that eqies sing the law of consevation of momentm in which it is impotant to know that momentm is a vecto qantity.

Poblems 183 Reading Passage Poblems BIO Heatbeat detecto A pisone ties to escape fom a Nashville, Tennessee pison by hiding in the landy tck. The pisone is spised when the tck is stopped at the gate. A gad entes the tck and handcffs him. How did yo know I was hee? the pisone asks. The heatbeat detecto, says the gad. A heatbeat detecto senses the tiny vibations cased by blood pmped fom the heat. With each heatbeat, blood is pmped pwad to the aota, and the body ecoils slightly, conseving the momentm of the blood-body system. The body s vibations ae tansfeed to the inside of the tck. Vibation sensos on the otside of the tck ae linked to a geophone, o signal amplifie, attached to a compte. A wave analyze pogam in the compte compaes vibation signals fom the tck to wavelets podced by heatbeat vibations. The wave analyze distingishes a peson s heatbeat fom othe vibations in the tck o in the sonding envionment, allowing gads to detect the pesence of a hman in the tck. 8. What does the heatbeat detecto sense? (a) Electic signals cased by electic dipole chages podced on the heat (b) Body vibations cased by blood pmped fom the heat (c) Sond cased by beathing (d) Slight ncontollable eflexive motions of an enclosed peson (e) All of the above 81. A heatbeat detecto elies on a geophone placed against the exteio of a tck o ca. What can case the vibations of the tck o ca? (a) Wind (b) Gond vibations de to othe moving cas o tcks (c) Vibations of passenges in the ca o tck (d) All of the above 82. What can be sed to analyze the motion of a peson s body hidden inside a ca o tck (choosing the body and blood as a system)? (a) The idea that mass of an isolated system is constant (b) The idea that momentm of an isolated system is constant (c) The implse-momentm pinciple (d) a and b (e) b and c 83. Ding each heatbeat, abot.8 kg of blood passes the aota in abot.16 s. This blood s velocity changes fom abot.8 m/s pwad towad the head to.8 m/s down towad the feet. What is the blood s acceleation? (a) zeo (b) 5 m>s 2 p (c) 5 m>s 2 down (d) 1 m>s 2 p (e) 1 m>s 2 down 84. Sppose.8 kg of blood moving pwad in the aota at.8 m/s eveses diection in.16 s when it eaches the aotic ach. If a pisone is tying to escape fom pison by hiding in a landy tck, and the mass of his body is 7 kg, which is the closest to the speed his body is moving immediately afte the blood changes diection passing thogh the aotic ach? (a).9 m/s (b).2 m/s (c).8 m/s (d).8 m/s (e).1 m/s Space Shttle lanch The mass of the Space Shttle at lanch is abot 2.1 * 1 6 kg. Mch of this mass is the fel sed to move the obite, which caies the astonats and vaios items in the shttle s payload. The Space Shttle geneally tavels fom 3.2 * 1 5 m (2 mi) to 6.2 * 1 5 m (385 mi) above Eath s sface. The shttle s two solid fel boostes (the cylindes on the sides of the shttle) povide 71.4% of the thst ding liftoff and the fist stage of ascent befoe being eleased fom the shttle 132 s afte lanch at 48,-m above sea level. The boostes contine moving p in fee fall to an altitde of appoximately 7, m and then fall towad the ocean to be ecoveed 23 km fom the lanch site. The shttle s five engines togethe povide 3.46 * 1 7 N of thst ding liftoff. 85. Which nmbe below is closest to the acceleation of the shttle ding liftoff? [Hint: Remembe the gavitational foce that Eath exets on the shttle.] (a) 3.3 m>s 2 (b) 6.6 m>s 2 (c) 9.8 m>s 2 (d) 16 m>s 2 (e) 33 m>s 2 86. Which nmbe below is closest to the aveage vetical acceleation of the shttle ding the fist 132 s of its flight? (a) 3.3 m>s 2 (b) 5.5 m>s 2 (c) 9.8 m>s 2 (d) 14 m>s 2 (e) 36 m>s 2 87. The boostes ae eleased fom the shttle 132 s afte lanch. How do thei vetical components of velocity compae to that of the shttle at the instant of elease? (a) The boostes vetical component of velocity is zeo. (b) The boostes vetical component of velocity is abot -9.8 m/s. (c) The vetical component of velocity of the boostes and that of the Shttle ae the same. (d) Thee is too little infomation to decide. 88. What is the appoximate implse of the jet engine thst exeted on the shttle ding the fist 1 s of flight? (a) 98 N # s downwad (b) 98 N # s pwad (c) 3.4 * 1 7 N # s pwad (d) 3.4 * 1 8 N # s pwad (e) 3.4 * 1 8 N # s downwad 89. What is the appoximate implse of Eath s gavitational foce exeted on the shttle ding the fist 1 s of flight? (a) 98 N # s downwad (b) 98 N # s pwad (c) 2.1 * 1 7 N # s pwad (d) 2.1 * 1 8 N # s pwad (e) 2.1 * 1 8 N # s downwad 9. What is the momentm of the Space Shttle 1 s afte liftoff closest to? (a) 2.1 * 1 6 kg # m>s down (b) 2.1 * 1 6 kg # m>s p (c) 2.1 * 1 7 kg # m>s p (d) 1.3 * 1 8 kg # m>s p (e) 1.3 * 1 8 kg # m>s down 91. What answe below is closest to the speed of the shttle and boostes when they ae eleased? Assme that the fee-fall gavitational acceleation at this elevation is abot 9.6 m>s 2 down. (a) 1 m/s (b) 3 m/s (c) 6 m/s (d) 1 m/s (e) 14 m/s