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# p0508 - 308 CHAPTER 5 IMPULSE AND MOMENTUM 5.8 Chapter 5...

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Unformatted text preview: 308 CHAPTER 5. IMPULSE AND MOMENTUM 5.8 Chapter 5, Problem 8 Problem: Two balls of mass m1 = m and m2 = 2m and coefficient of restitution e approach each other with velocities v1 = V i and v2 = − 3 V i. Determine their velocities after the impact, v1 and v2 , 2 in terms of V and e. If v2 = 0, what are e and v1 ? .... .... ....... ...... . .......................... .. . ..... ............................. .................................. .. ...................................... ..................... . 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 = − 3 V , while m1 = m and m2 = 2m, momentum conservation along the line of impact tells us 2 that 3 mV − 2m V = mv1 + 2mv2 2 Dividing through by m yields v1 + 2v2 = −2V Impact Relation. For a coefficient of restitution equal to e, we have v2 − v1 = e (v1 − v2 ) = 5 v2 = v1 + eV 2 Substituting for v2 in the momentum equation tells us that 5 v1 + 2 v1 + eV 2 Again using the impact relation, we have 1 5 v2 = − (5e + 2)V + eV 3 2 Therefore, the velocity vectors after the impact are v1 v2 For the special case v2 = 0, we have 1 (5e − 4)V = 0 6 Solving for v1 yields v1 = − 1 3 4 5 + 2 V i = −2V i 5 =⇒ e= 4 5 1 = − (5e + 2)V i 3 1 = (5e − 4)V i 6 =⇒ v2 = 1 (5e − 4)V 6 = −2V =⇒ 1 v1 = − (5e + 2)V 3 5 eV 2 To complete the solution, we solve for v2 in terms of v1 from the impact relation, which yields ...
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