5.4 Head-on Elastic Collisions

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1 5.4 Head-on Elastc Collsons In prevous sectons, you read about systems nvolvng elastc and nelastc collsons. You appled the law of conservaton of momentum and the law of conservaton of netc energy to solve problems. You also read about the deal cases of the perfectly elastc and perfectly nelastc collsons, and learned that n both cases momentum s conserved. You dscovered that netc energy s conserved n the case of perfectly elastc collsons, but not nelastc collsons. In ths secton, you wll examne n greater detal stuatons nvolvng perfectly elastc collsons n one dmenson. You wll calculate the fnal veloctes of two objects after an elastc collson n one dmenson wth equatons for specal cases. Once agan, we consder near-perfectly elastc collsons n one dmenson (Fgure ). Fgure Glders on an ar trac can undergo elastc collsons n one dmenson. head-on elastc collson an mpact n whch two objects approach each other from opposte drectons; momentum and netc energy are conserved after the collson Perfectly Elastc Head-on Collsons n One Dmenson In a one-dmensonal head-on elastc collson, two objects approach each other from opposte drectons and collde. In such collsons, both momentum and netc energy are conserved. You can derve expressons for the fnal veloctes of two objects n a head-on collson n terms of the ntal veloctes and the objects masses. Suppose an object of mass travels wth ntal velocty v and colldes head-on wth an object of mass m travellng at velocty v. If we assume a one-dmensonal collson, we can omt the vector notaton for veloctes, and nstead use postve and negatve values to dentfy moton n one drecton or the opposte drecton. We begn the analyss wth the conservaton of momentum: m v 5 v f m v f (Equaton ) Rewrte Equaton by brngng all the terms wth to one sde and all the terms wth m to the other sde, and common factorng the m coeffcents: v f 5 m v f m v v v f 5 m v f v (Equaton ) Snce ths s an elastc collson, conservaton of total netc energy can be appled: v m v 5 v f m v f Multply both sdes of the equaton by to clear the fractons: a v m v b 5 a v f m v f b v m v 5 v f m v f Collect terms on the left sde and m terms on the rght sde and dvde out the common factors. v v f 5 m v f m v v v f 5 m v f v Factor both sdes usng the dfference of squares: v v f v 5 m v f v v f v (Equaton 3) Dvde Equaton 3 by Equaton : v v f v 5 m v f v v f v v v f m v f v v 5 v f v (Equaton 4) 40 Chapter 5 Momentum and Collsons

2 Rearrangng Equaton 4 to solate v f on the left gves v f 5 v v (Equaton 5) Substtute Equaton 5 nto Equaton to express v f n terms of the masses and ther ntal speeds: m v 5 v f m v v m v 5 v f m v m v f m v Collect the v f terms on the rght sde of the equaton and terms nvolvng ntal veloctes on the left sde, then collect le terms and dvde out common m coeffcents: m v m v m v 5 v f m v f m v m v 5 m v f Dvde both sdes by m to solate v f : 5 a m b v > m a b v > m Ths equaton expresses the fnal velocty of the frst object n terms of the masses and ntal veloctes of the two objects. Smlarly, we can rearrange Equaton 4 to solate v f on the left: v f 5 v f v v (Equaton 6) To derve a smlar equaton for v f, follow the above steps for v f, startng wth substtutng Equaton 6 nto Equaton, and endng by dvdng both sdes by m and solatng v f on the left sde: v > f 5 a m bv > m a bv > Ths equaton expresses the fnal velocty of the second object n terms of the masses and ntal veloctes of the two objects. Note that the equaton for v > f s the same as the equaton for f you nterchange all the and subscrpts. It s mportant to note that these equatons hold true only for perfectly elastc collsons n one dmenson. In some cases, one of the objects s ntally at rest. For nstance, f v s ntally zero, the equatons above smplfy to 5 a m m bv > v > f 5 a bv > In Tutoral, you wll use these velocty relatonshps as an alternatve way of analyzng some head-on elastc collsons. As you do Tutoral, compare these methods to the methods used n Secton 5.3. WEB LINK 5.4 Head-on Elastc Collsons 4

3 Tutoral Analyzng Head-on Elastc Collsons In these Sample Problems, you wll use the relatve velocty relatonshps derved above to solve problems related to head-on elastc collsons n one dmenson. Sample Proble: Head-on Elastc Collson wth One Object at Rest n One Dmenson Consder an elastc head-on collson between two balls of dfferent masses, as shown n Fgure. The mass of ball s. g, and ts velocty s 7. m/s [W]. The mass of ball s 3.6 g, and ball s ntally at rest. Determne the fnal velocty of each ball after the collson. v 0 m/s m Fgure v 7. m/s [W] Gven: 5. g; v > 5 7. m/s 3W4; m g; v > 5 0 m/s Requred: ; v > f Analyss: Snce one of the objects s ntally at rest n the head-on elastc collson, use the smplfed equatons: 5 a m m bv > v > f 5 a bv > Soluton: Let the negatve x-drecton represent west. For ball, 5 a m m bv >. g 3.6 g 5 a. g 3.6 g b 7. m/s m/s m/s 3E4 For ball, v > f 5 a bv >. g 5 a. g 3.6 g b 7. m/s m/s v > f m/s 3W4 Statement: The fnal velocty of ball s 3.6 m/s [E]. The fnal velocty of ball s 3.6 m/s [W]. Sample Problem : Head-on Elastc Collson wth Both Objects Movng n One Dmenson In a bumper car rde, bumper car has a total mass of 350 g and s ntally movng at 4.0 m/s [E]. In a head-on completely elastc collson, bumper car hts bumper car. The total mass of bumper car s 50 g, and t s movng at.0 m/s [W]. Calculate the fnal velocty of each bumper car mmedately after the collson. Gven: Let east be postve and west be negatve; g; v > m/s [E] m/s; m 5 50 g; v > 5.0 m/s [W] 5.0 m/s Requred: ; v > f Analyss: 5 a m bv > m a bv > m v > f 5 a m bv > m a bv > Soluton: 5 a m bv > m a bv > m 350 g 50 g 5 a 350 g 50 g b 4.0 m/s 50 g a 350 g 50 g b.0 m/s 5.0 m/s 5.0 m/s 3W4 v > f 5 a m bv > m a bv > 50 g 350 g 5 a 350 g 50 g b.0 m/s 350 g a 350 g 50 g b 4.0 m/s m/s v > f m/s 3E4 Statement: The fnal velocty of bumper car s.0 m/s [W]. The fnal velocty of bumper car s 5.0 m/s [E]. 4 Chapter 5 Momentum and Collsons

4 Practce. A ball of mass 80.0 g s movng at 7.0 m/s [W] when t undergoes a head-on elastc collson wth a statonary ball of mass 60.0 g. Assume the collson s one-dmensonal. Calculate the velocty of each ball after the collson. T/I [ans:.0 m/s [W]; 8.0 m/s [W]]. Cart has a mass of.5 g and s movng on a trac at 36.5 cm/s [E] toward cart. The mass of cart s 5 g, and t s movng toward cart at 4.8 cm/s [W]. The carts collde. The collson s cushoned by a Hooe s law sprng, mang t an elastc head-on collson. Calculate the fnal velocty of each cart after the collson. T/I [ans: cart : 90 cm/s [W]; cart : 6 cm/s [W]] Specal Cases Usng these new equatons for head-on elastc collsons n one dmenson, specal cases of collsons, such as objects of equal mass, produce some nterestng results. Case : Objects Have the Same Mass The frst case we consder s when the objects that are colldng have the same mass, so let 5 m 5 m. 5 a m m m bv> a m m bv> 5 a 0 m bv> a m m bv> 5 a m m bv> 5 v > v > f 5 a m m m bv> a m m bv> 5 a 0 m bv> a m m bv> 5 a m m bv> v > f 5 v > In other words, when two objects wth the same mass undergo a head-on elastc collson n one dmenson, they exchange veloctes almost as f they pass through each other. Case : A Lghter Object Colldng wth a Much Heaver, Statonary Object Our second case deals wth stuatons n whch the mass of one of the objects s much greater than the mass of the other object, and the heaver object s statonary. For example, f object s statonary and has a much greater mass, then snce m s much greater than, you can consder to be approxmately zero, or neglgble. So 5 a m m bv > L a 0 m bv > 0 L v > v > f 5 a bv > L a 0 0 m bv > v > f L Head-on Elastc Collsons 43

5 Investgaton 5.4. Head-on Elastc Collsons (page 59) Now that you have an understandng of how head-on elastc collsons wor, perform Investgaton 5.4. to study these types of collsons n greater detal. In other words, f an object colldes wth a statonary, much heaver object, the velocty of the lght object s reversed, and the heaver object stays at rest. To put ths scenaro nto perspectve, consder a collson between a table tenns ball and a statonary transport truc: the transport truc wll not move and the table tenns ball wll bounce bac wth the same speed. You wll explore more specal cases n the questons at the end of Secton 5.4. Conservaton of Mechancal Energy You have dscovered what happens to momentum n head-on elastc collsons. What do you suppose happens to the conservaton of total mechancal energy durng elastc collsons? One of the two glders n Fgure 3 has been ftted wth a sprng bumper. When the two glders collde head-on n an elastc collson, the bounce s not mmedate. If you vewed the collson n slow moton, you would see the bumper compress ntally and then sprng bac to ts orgnal shape. Durng the compresson, some of the netc energy of the movng glders s converted nto elastc potental energy. Ths potental energy s converted bac nto netc energy durng the rebound. sprng bumper v v Fgure 3 When the two glders collde, the duraton of the collson s greater than t would be wthout the sprng bumper on one of the glders. If the compresson of the sprng bumper durng the collson s x, then the law of conservaton of energy states: v m v 5 v f m v f x Ths equaton and the graph n Fgure 4 both show that as the sprng compresses, the elastc energy ncreases and the total netc energy of the two carts decreases. The total mechancal energy, however, stays constant. As the compresson decreases, the elastc energy decreases and the total netc energy ncreases. The total mechancal energy stll remans constant. Mechancal energy (J) total mechancal energy total netc energy total elastc potental energy 0 Tme (s) Fgure 4 In ths graph of total mechancal energy versus tme, you can see how the total mechancal energy, the total netc energy, and the total elastc potental energy relate to each other throughout the collson. 44 Chapter 5 Momentum and Collsons

6 To determne the maxmum compresson of the sprng durng the collson, use the fact that when the two glders collde they have the same velocty at that pont. If they dd not have the same velocty at maxmum compresson, then one would be catchng up to the other or pullng away from the other. Therefore, at maxmum compresson (closest approach), the two objects must have the same velocty, v f. The equaton above then reduces to the followng: v m v 5 m v f x In Tutoral, you wll apply the conservaton of mechancal energy to problems nvolvng the physcs of sprng carts. Tutoral Applyng Conservaton of Mechancal Energy In the followng Sample Problem, you wll apply the conservaton of mechancal energy to solve collson problems. Sample Proble: Two-Cart Sprng System Dynamcs cart has a mass of.8 g and s movng wth a velocty of 4.0 m/s [rght] along a frctonless trac. Dynamcs cart has a mass of. g and s movng at 6.0 m/s [left]. The carts collde n a head-on elastc collson cushoned by a sprng wth sprng constant N/m (Fgure 5). (a) Determne the compresson of the sprng, n centmetres, durng the collson when cart s movng at 4.0 m/s [left]. (b) Calculate the maxmum compresson of the sprng, n centmetres. Fgure 5 cart cart Soluton (a) Gven: 5.8 g; v > m/s 3rght4; m 5. g; v > m/s 3left4; v > f m/s 3left4; N/m Requred: x Analyss: Use the conservaton of momentum to determne the velocty of cart durng the collson, when cart s movng 4.0 m/s [left]. Then apply the conservaton of mechancal energy to determne the compresson of the sprng at ths partcular moment durng the collson. Consder rght to be postve and left to be negatve, and omt the vector notaton. Soluton: Begn wth the conservaton of momentum equaton. m v 5 v f m v f Rearrange ths equaton to express the fnal velocty of cart n terms of the other gven values. m v m v f 5 v f x m v m v f 5 v f 5.4 Head-on Elastc Collsons 45

7 Substtute the gven values and solve. v f 5 v m v m v f.8 g 4.0 m/s. g 6.0 m/s. g 4.0 m/s 5.8 g v f 5.56 m/s one extra dgt carred Cart s movng.6 m/s [rght] when cart s movng 4.0 m/s [left]. Now use the conservaton of mechancal energy to determne the compresson of the sprng, x. v m v 5 v f m v f x Multply both sdes of the equaton by to clear the fractons, and then solate the term contanng x on one sde of the equaton. a v m v v f m v f b 5 a x b Dvde both sdes by and then tae the square root of both sdes. v m v v f m v f 5 x Å v m v v f m v f 5 x Substtute the nown values to determne the compresson of the sprng when cart s movng 4.0 m/s [left]. x 5 Å v m v v f m v f x 5 Å.8 g 4.0 m/s. g 6.0 m/s.8 g.56 m/s. g 4.0 m/s N/m x m Statement: The compresson of the sprng s.9 cm durng the collson, when cart s movng 4.0 m/s [left]. (b) Gven: 5.8 g; v > m/s 3rght4; m 5. g; v > m/s 3left4; N/m Requred: x Analyss: At the begnnng of the collson, as the carts come together and the sprng s beng compressed, cart s movng faster than cart. Toward the end of the collson, as the carts separate and the sprng s beng released, cart wll be movng faster than cart. At the pont of maxmum compresson of the sprng, the two carts wll have the same velocty, v f. Use the conservaton of momentum equaton to determne ths velocty. Then apply the conservaton of mechancal energy to calculate the maxmum compresson of the sprng. Soluton: m v 5 v f m v f Factor out the common factor v f. m v 5 m v f 46 Chapter 5 Momentum and Collsons

8 Dvde both sdes by m to solate v f. m v 5 v m f Substtute the gven values to calculate the velocty of both carts at maxmum compresson. v f 5 v m v m.8 g 4.0 m/s. g 6.0 m/s 5.8 g. g v f 5.5 m/s Now use the law of conservaton of mechancal energy to determne the maxmum compresson of the sprng. Clear the fractons frst, and then solate x. a v m v b 5 a m v f x b v m v m v f 5 x v m v m v f 5 x Å Substtute the nown values and solve for the maxmum compresson of the sprng. x 5 Å v m v m v f 5 Å.8 g 4.0 m/s. g 6.0 m/s.8 g. g.5 m/s N/m x m Statement: The maxmum compresson of the sprng s 3.5 cm. Unt TASK BOOKMARK You can apply what you learn about head-on elastc collsons and conservaton of mechancal energy to the Unt Tas on page 70. Practce. A. g glder movng at 3.0 m/s [rght] undergoes an elastc head-on collson wth a glder of equal mass movng at 3.0 m/s [left]. The collson s cushoned by a sprng whose sprng constant,, s N/m. T/I A (a) Determne the compresson n the sprng when the second glder s movng at.5 m/s [rght]. [ans:.6 cm] (b) Calculate the maxmum compresson of the sprng. [ans:.9 cm]. A student desgns a new amusement par rde that nvolves a type of bumper car that has a sprng on the front to cushon collsons. To test the bumper, the student attaches one sprng to the front of a sngle car. In a collson, car wth total mass g s movng at 3.0 m/s [E] toward car wth total mass g movng at 3.3 m/s [W]. Durng the collson, the sprng compresses a maxmum of 44 cm. Determne the sprng constant. T/I [ans: N/m] 5.4 Head-on Elastc Collsons 47

9 5.4 Revew Summary In a perfectly elastc head-on collson n one dmenson, momentum and netc energy are conserved. Usng the law of conservaton of momentum and the law of conservaton of netc energy, we can derve equatons to determne the fnal veloctes of two objects n a perfectly elastc head-on collson n one dmenson: 5 m m v > m m v > and v > f 5 m m v > m m v >. In cases where v > s ntally zero, 5 m m v > and v > m f 5 m v >. In cases where the masses of the colldng objects are dentcal, 5 v > and v > f 5 v >. In cases n whch one mass s sgnfcantly larger than the other mass, and the larger mass s statonary, L v > and v > f L 0. Durng a head-on collson n one dmenson, the netc energy of the movng masses s converted nto elastc potental energy, and then bac nto netc energy durng the rebound. Total mechancal energy s conserved throughout the collson. Questons. Is t possble for two movng masses to undergo an elastc head-on collson and both be at rest mmedately after the collson? Is t possble for an nelastc collson? Explan your reasonng. K/U. In curlng, you wll often see one curlng stone ht another and come to rest whle the statonary stone moves away from the one-dmensonal collson. Explan how ths can happen. K/U 3. The partcles n Fgure 6 undergo an elastc collson n one dmenson. Partcle has mass.5 g and partcle has mass 3.5 g. Ther veloctes before the collson are v > 5 m/s 3rght4 and v > m/s 3left4. Determne the velocty of the two partcles after the collson. K/U T/I A m x Fgure 6 4. Two chuns of space debrs collde head-on n an elastc collson. One pece of debrs has a mass of.67 g. The other chun has a mass of 5.83 g. After the collson, both chuns move n the drecton of the second chun s ntal velocty wth speeds of 85 m/s for the smaller chun and 7 m/s for the larger. What are the ntal veloctes of the two chuns? K/U T/I 5. Dynamcs cart has a mass of 0.84 g and s ntally movng at 4. m/s [rght]. Cart undergoes an elastc head-on collson wth dynamcs cart. The mass of cart s 0.48 g, and cart s ntally movng at.4 m/s [left]. The collson s cushoned by a sprng wth sprng constant N/m. T/I (a) Calculate the fnal velocty of each cart after they completely separate. (b) Determne the compresson of the sprng durng the collson at the moment when cart s movng at 3.0 m/s [rght]. (c) Determne the maxmum compresson of the sprng. 6. Ball has a mass of.0 g and s suspended wth a 3.0 m rope from a post so that the ball s statonary. Ball has a mass of 4.0 g and s ted to another rope. The second rope also measures 3.0 m but s held at a 60.0 angle, as shown n Fgure 7. When ball s released, t colldes, head-on, wth ball n an elastc collson. T/I (a) Calculate the speed of each ball mmedately after the frst collson. (b) Calculate the maxmum heght of each ball after the frst collson. 4.0 g Fgure m g 3.0 m 48 Chapter 5 Momentum and Collsons

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