Physics 100A Homework 8 Chapter 9

Transcription

1 Physcs 00A Hoework 8 Chater Two ar-track carts oe toward one another on an ar track. Cart has a ass o 0.35 kg and a seed o. /s. Cart has a ass o 0.6 kg. A)What seed ust cart hae the total oentu o the syste s to be zero? B)Snce the oentu o the syste s zero, does t ollow that the knetc energy o the syste s also zero? 4. Pcture the Proble: The two carts aroach each other on a rctonless track at derent seeds. Strategy: Add the oenta o the two carts and set t equal to zero. Sole the resultng exresson or. Then use equaton 7-6 to nd the total knetc energy o the two-cart syste. Let cart trael n the oste drecton. Soluton:. (a) Set 0 and sole or : + 0 Ths s a one-densonal roble so the arrow s ( 0.35 kg)(. /s ) 0.69 /s droed but the drecton s taken nto consderaton a 0.6 kg the lus/nus sgn. The absolute o the quantty s used to calculate the seed, whch s a scalar.. (b) No, knetc energy s always greater than or equal to zero. 3. (c) Use equaton 7-6 to su the knetc K + energes o the two carts: 0.35 kg. /s kg 0.69 /s 0.40 J Insght: I cart s traelng n the oste x drecton, then ts oentu s cart s ( 0.4 kg /s)ˆ x. ( )( ) ( )( ) ˆ ( ) ˆ 0.4 kg /s x and the oentu o 9.7 Object has a ass and a elocty.0 /s xˆ. Object has a ass and a elocty ˆ 3.50 /s y. The total oentu o these two objects has a agntude o 7.6 kg /s and onts n a drecton 66.5 aboe the oste x axs. 7. Pcture the Proble: The nddual oenta and nal oentu ectors are dected at rght. Strategy: The oenta o the two objects are erendcular. Because o ths we can say that the oentu o object s equal to the x-coonent o the total oentu and the oentu o object s equal to the y-coonent o the total oentu. Fnd the oenta o objects and n ths anner and dde by ther seeds to deterne the asses. Soluton:. Fnd total, x and dde by : total, x total θ ( )( ) cos 7.6 kg /s cos kg / 7.0 kg /s.80 /s.5 kg y x total. Fnd total, y and dde by : total, y total 3.0 /s ( )( ) snθ 7.6 kg /s sn kg /s 6. kg /s 5. kg Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9

2 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton Insght: Note that object has the larger oentu because the total oentu onts ostly n the ŷ drecton. The two objects hae slar seeds, so object ust hae the larger ass n order to hae the larger oentu. Iulse on a Ball In a baseball gae the batter swngs and gets a good sold ht. Hs swng ales a orce o,000 N to the ball or a te o s. A) Assung that ths orce s constant, what s the agntude J o the ulse on the ball? 3 J FΔ t (,000)(0.7 0 ) 8.4 kg /s B) The net orce ersus te grah has a rectangular shae. Oten n hyscs geoetrc roertes o grahs hae hyscal eanng. For ths grahs the area o the rectangle corresonds to the ulse. C) I both the grah reresentng the constant net orce and the grah reresentng the arable net orce reresent the sae ulse actng on the baseball, the two grahs ust hae the sae area. D) Assue that a tcher throws a baseball so that t traels n a straght lne arallel to the ground. The batter then hts the ball so t goes drectly back to the tcher along the sae straght lne. Dene the drecton the tcher orgnally throws the ball as the +x drecton. The ulse on the ball caused by the bat wll be n the negate x-drecton. J J ( ˆ) ˆ ball x ballx J ( + ) x ˆ ball E) Now assue that the tcher n Part D throws a 0.45-kg baseball arallel to the ground wth a seed o 3 /s n the +x drecton. The batter then hts the ball so t goes drectly back to the tcher along the sae straght lne. What s the ball's elocty just ater leang the bat the bat ales an ulse o -8.4 Ns to the baseball? J ball J ( 84) ball /s (to the let) Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9

3 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton Knetc Energy and Moentu The two toy cars shown n the gure, wth asses as gen n the gure, are ready to race. Both cars begn ro rest. For each queston, state whether the correct answer s car A, car B, or whether the two cars hae equal alues or the araeter n queston. For the next three arts assue that the cars' otors suly the sae orce to each car oer the course o a.0- eter race. A)Whch car crosses the nsh lne.0 away rst? Car B wns. 0 or both cars ntal F a Then. Snce the orce s the sae the car wth the sallest ass wll hae the larger acceleraton and wll coer the.0 dstance aster. B) Whch car has the larger knetc energy when t crosses the nsh lne.0 away? The sae. W Fd Δ K K - K K Snce the orce and dslaceent are the sae the cars wll nal ntal nal hae the sae nal knetc energy. C) Whch car has a larger oentu when t crosses the nsh lne.0 away? Fro the knetc energy ( W) / nal And W Snce the work s the sae the car wth the largest ass wll hae the nal nal larger oentu. D) Whch car has traeled arther ater 0s? Car B wth the larger acceleraton. E) Ater 0 s whch car has a larger knetc energy? Car B would hae traelled the largest dstance and thereore exerenced the largest work whch corresonds to a hgher nal knetc energy. F) Ater 0 s whch car has a larger oentu? $_{nal} F \Delta t$ I the orce and the te are the sae, the nal oentu s the sae or both. 9.9) A 0.4-kg baseball oes toward hoe late wth a elocty ( 36/s) ˆ x Ater strkng the bat, the ball oes ertcally uward wth a elocty (8 /s ) yˆ. A) Fnd the drecton o the ulse delered to the ball by the bat. Assue that the ball and bat are n contact or.5s. Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9 3

4 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton B)Fnd the agntude o the ulse delered to the ball by the bat. Assue that the ball and bat are n contact or.5s. 9. Pcture the Proble: The ball rebounds ro the bat n the anner ndcated by the gure at rght. Strategy: The ulse s equal to the ector change n the oentu. Analyze the x and y coonents o Δ searately, then use the coonents to nd the drecton and agntude o I. Soluton:. (a) Fnd Δ : ( ) ( ) ( ) x Δ x x x 0.4 kg 0 36 /s 5.0 kg /s. Fnd y : Δ ( ) ( )( ) 3. Use equaton 9-6 to nd I: Δ y y y 0.4 kg 8 0 /s.5 kg /s I Δ x+ ( 5.0 kg /s) ˆ (.5 kg /s) I y.5 4. Fnd the drecton o I: θ tan tan 7 aboe the horzontal I x Fnd the agntude o I: x y yˆ ( ) ( ) I I + I 5.0 kg /s +.5 kg /s 5.6 kg /s 6. (b) I the ass o the ball were doubled the ulse would double n agntude. There would be no change n the drecton. 7. (c) I Δ o the ball s unchanged, the ulse delered to the ball would not change, regardless o the ass o the bat. Insght: The ulse brngs the ball to rest horzontally but ges t an ntal horzontal seed. Very or yoursel that ths ball wll trael straght uward 6.5 (54 eet) beore allng back to Earth. An easy ou! Moentu n an Exloson A gant "egg" exlodes as art o a reworks dslay. The egg s at rest beore the exloson, and ater the exloson, t breaks nto two eces, wth the asses ndcated n the dagra, traelng n ooste drectons A)What s the oentu A, o ece A beore the exloson? Snce the egg s ntally at rest the ntal oentu s zero or both eces. B) Durng the exloson, s the orce o ece A on ece B greater than, less than, or equal to the orce o ece B on ece A? By Newton s thrd law the orce o A on B has the sae agntude but ooste drecton to the orce o B on A. C)The oentu o ece B s easured to be 500 kg /s ater the exloson. Fnd the oentu A, o ece A ater the exloson. + A, B, 500 kg /s. total, total, 0 A, B, 0 Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9 4

5 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton Catchng a Ball on Ice Ola s standng on a sheet o ce that coers the ootball stadu arkng lot n Bualo, New York; there s neglgble rcton between hs eet and the ce. A rend throws Ola a ball o ass kg that s traelng horzontally at.8 /s. Ola's ass s 70.6 kg. A)I Ola catches the ball, wth what seed do Ola and the ball oe aterward? O, + b, ( O + b) b, Stck together b b, (0.4)(.8) 6.65 ( + ) ( ) c/s O b B) the ball hts Ola and bounces o hs chest horzontally at 7.0 /s the ooste drecton, what s hs seed ater the collson? O, + b, O, + b, O, b, b, b( b, b, ) b 0.4 O, ( b, b, ) (.8 ( 7.)) 0.7 c/s, n the orgnal drecton o the ball 70.6 O 9.) Two grous o canoests eet n the ddle o a lake. Ater a bre st, a erson n canoe ushes on canoe to searate the canoes. Suose the seeds o the two canoes ater they are ushed aart are 0.58 /s or canoe and 0.4 /s or canoe. I the ass o canoe s 30 kg, what s the ass o canoe?. Pcture the Proble: The two canoes are ushed aart by the orce exerted by a assenger. Strategy: By alyng the conseraton o oentu we conclude that the total oentu o the two canoes ater the ush s zero, just as t was beore the ush. Set the total oentu o the syste to zero and sole or. Let the elocty ont n the negate drecton, n the oste drecton. Soluton: Set total 0 and sole or : x x x x ( 30 kg)( 0.58 /s) x x 0.4 /s 440 kg Insght: An alternate way to nd the ass s to use the equatons o kneatcs n a anner slar to that descrbed n Exale 9-3. Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9 5

6 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton 9.5) A 9-kg astronaut and a 00-kg satellte are at rest relate to the sace shuttle. The astronaut ushes on the satellte, gng t a seed o 0.4 /s drectly away ro the shuttle. Seen-and-a-hal seconds later the astronaut coes nto contact wth the shuttle. What was the ntal dstance ro the shuttle to the astronaut? 5. Pcture the Proble: The astronaut and the satellte oe n ooste drectons ater the astronaut ushes o. The astronaut traels at constant seed a dstance d beore cong n contact wth the sace shuttle. Strategy: As long as there s no rcton the total oentu o the astronaut and the satellte ust rean zero, as t was beore the astronaut ushed o. Use the conseraton o oentu to deterne the seed o the astronaut, and then ultly the seed by the te to nd the dstance. Assue the satellte s oton s n the negate x-drecton. Soluton:. Fnd the seed o the astronaut usng conseraton o oentu:. Fnd the dstance to the sace shuttle: a + s 0 aa + ss s s a s s d at t a a ( 00 kg)( 0.4 /s) ( ) ( 9 kg) 7.5 s 4 Insght: One o the trcky thngs about sacewalkng s that wheneer you ush on a satellte or anythng else, because o Newton s Thrd Law you yoursel get ushed! Conseraton o oentu akes t easy to redct your seed. A Grl n a Traolne h.0 5 A grl o ass klogras srngs ro a traolne wth an ntal uward elocty o eters er second. At heght eters aboe the traolne, the grl grabs a box o ass klogras. A) What s the seed beore o the grl edately beore she grabs the box? Here we use kneatcs. gy beore beore (80) (98)() 4.98 /s B) What s the seed $_{ater}$ o the grl edately ater she grabs the box? Now we use conseraton o oentu. bb, + gg, beore, ( g + b) ater, b, 0 g g, beore (60)(498) ater 3.98 ( + ) (60+ 5) /s g b C) Ths "collson" s nelastc. D) What s the axu heght that the grl (wth box) reaches? Measure wth resect to the to o the h ax traolne. Let us start at the leel n whch the grl grabs the box and then add the addtonal heght. gy, the nal elocty s zero as the grl and box reach the axu heght. (398) y ater 0.8 g g (98) h And ax h ax Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9 6

7 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton A One-Densonal Inelastc Collson Block, o ass 8.70 kg, oes along a rctonless ar track wth seed 5.0 /s. It colldes wth block, o ass 3.0 kg, whch was ntally at rest. The blocks stck together ater the collson. A) Fnd the agntude o the total ntal oentu o the two-block syste., +, (87)(5) Kg /s B) Fnd, the agntude o the nal elocty o the two-block syste. 3 total 4. /s total ( ) C) What s the change Δ K K K n the syste's knetc energy due to the collson? Δ K total Δ K ( )(4.) (8.7)(5) 0 70 J 9.8) A cart o ass oes wth a seed on a rctonless ar track and colldes wth an dentcal cart that s statonary. I the two carts stck together ater the collson, what s the nal knetc energy o the syste? total ( + ) ( ) 0.5 total K 8. Pcture the Proble: The two carts collde on a rctonless track and stck together. Strategy: The collson s coletely nelastc because the two carts stck together. Moentu s consered durng the collson because the track has no rcton. The two carts oe as they were a sngle object ater the collson. Use the conseraton o oentu to nd the nal seed o the carts and nal knetc energy o the syste. Soluton:. Consere oentu to nd the nal seed o the carts: + (0). Use equaton 7-6 to nd the nal knetc energy: ( ) K ( ) 4 Insght: Hal o the ntal knetc energy s gone, hang been conerted to heat, sound, and eranent deoraton o ateral durng the nelastc collson. Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9 7

8 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton 9.37) A 73-kg car stoed at an ntersecton s rear-ended by a 70-kg truck ong wth a seed o 5.5 /s. A) I the car was n neutral and ts brakes were o, so that the collson s aroxately elastc, nd the nal seed o the truck. B) Fnd the nal seed o the car. Let us dere the equatons ro the book. Moentu conseraton wth,, 0 and, 0, 0 + Equaton ( ) Equaton, 0, Knetc energy conseraton, +, 0 ( ), 0, ( )( + ) Equaton 3, 0, 0, Substtutng ro equaton nto equaton 3 ( + ),, 0, + Equaton 4, 0, Substtute nto equaton + ( + ),, 0 0 Collectng ters ( + ), ( ) 0 ( ), 0 ( + ) And returnng to equaton 4, 0 ( + ) 37. Pcture the Proble: The truck strkes the car ro behnd. The collson sends the car lurchng orward and slows down the seed o the truck. Strategy: Ths s a one-densonal, elastc collson where one o the objects (the car) s ntally at rest. Thereore, equaton 9- ales and can be used to nd the nal seeds o the ehcles. Let be the ass o the truck, be the ass o the car, and be the ntal seed o the truck. Soluton:. Use equaton 9- to nd :, 0. Use equaton 9- to nd, : kg 5.5 /s 6.5 /s ( ), 0 truck kg ( ) 70 kg ( 5.5 /s).7 /s, 0 car kg Insght: The elastc collson roduces a bgger jolt or the car. I the collson were nstead nelastc and the two ehcles stuck together, the nal seed o the car (and the truck) would be 0.9 /s. Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9 8

9 Chater 9: Lnear Moentu and Collsons Jaes S. Walker, Physcs, 4 th Edton 9.4) The three ar carts shown n the gure hae asses, readng ro let to rght, o 4,, and, resectely. The ost asse cart has an ntal seed o equed wth srng buers that ge elastc collsons. 0 ; the other two carts are at rest ntally. All carts are A) Fnd the nal seed o each cart. (Assue the ar track s long enough to accoodate all collsons.) We use the sae equatons as n the last roble: 4. Pcture the Proble: The cart 4 colldes wth the cart, whch s gen knetc energy as a result and later colldes wth the cart. Strategy: In each case a ong cart colldes wth a cart that s at rest, so alcaton o equaton 9- wll yeld the nal eloctes o all the carts. Frst aly equaton 9- to the collson between carts 4 and, then to the collson between and. Let the 4 cart be called cart 4, the cart be called cart, and the cart be called cart : Soluton:. (a) Aly equaton 9- to the rst collson: 4 4 4, 4, ( ) 4 4 4, 4, Aly equaton 9- to the second collson. In ths case cart has an ntal seed o (b) Very that K K by wrtng usng equaton 7-6 and ddng both sdes by 0 : : ( ) ( ) 4 ( ) 4 4,, ,, ? ( 4 ) ( 4)( ) + ( )( ) + ( )( ) Insght: Note that due to the transer o knetc energy a collsons, the cart wth the sallest ass ends u wth the largest seed. 0 Coyrght 00 Pearson Educaton, Inc. All rghts resered. Ths ateral s rotected under all coyrght laws as they currently exst. No orton o ths ateral ay be reroduced, n any or or by any eans, wthout ersson n wrtng ro the ublsher. 9 9