p0506 - 306 CHAPTER 5. IMPULSE AND MOMENTUM 5.6 Chapter 5,...

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Unformatted text preview: 306 CHAPTER 5. IMPULSE AND MOMENTUM 5.6 Chapter 5, Problem 6 Problem: Two balls of mass m1 = m and m2 = 9m and coefficient of restitution e approach each other with velocities v1 = V i and v2 = −V i. Determine their velocities after the impact, v1 and v2 , in terms of V and e. If |v1 | = 3|v2 |, what is e? ... ..... ... ............... .... ................... .............................. .................................. . .. .. ....................................... ....................................... 1 ......................................... .. ........................................... . ............................................................... .. ................................................................. . ........................................... . . . ........................................... . .. . ........................................... .. ............1..... .. .......................................... ....................... . ...................................... .................................... . .... . .... .............................. .................... ............. .......... . . .... m v .... ... ... .. ............... ......................... ............................. .................................. .. ................... ........................ 2 ........................................... . ...................................... ... ............ ... .......... . ............................................... . . .................................................................. ........................................................................ .. .. . ...... . . ..... ... . . . .........................2............. ..... ............. ..... ........................................... .......................................... ..... . .. . ..................................... ................................... . ................................ ...................... ................ ......... ... v m Solution: This is a direct central impact with the line of impact for the balls being the x axis. Momentum Conservation. Using the fact that the initial velocity components for Balls 1 and 2 are v1 = V and v2 = −V , while m1 = m and m2 = 9m, momentum conservation along the line of impact tells us that mV − 9mV = mv1 + 9mv2 Dividing through by m yields v1 + 9v2 = −8V Impact Relation. For a coefficient of restitution equal to e, we have v2 − v1 = e (v1 − v2 ) = 2eV To complete the solution, we solve for v2 in terms of v1 from the impact relation, which yields v2 = v1 + 2eV Substituting for v2 in the momentum equation tells us that v1 + 9 (v1 + 2eV ) = −8V Again using the impact relation, we have 1 v2 = − (4 + 9e)V + 2eV 5 Therefore, the velocity vectors after the impact are v1 v2 For the special case |v1 | = 3|v2 |, we have 1 3 (4 + 9e)V = (4 − e)V 5 5 Solving for e yields e= 2 3 =⇒ 4 + 9e = 12 − 3e 1 = − (4 + 9e)V i 5 1 = − (4 − e)V i 5 =⇒ 1 v2 = − (4 − e)V 5 =⇒ 1 v1 = − (4 + 9e)V 5 ...
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