p0506 - 234 CHAPTER 5. IMPULSE AND MOMENTUM 5.6 Chapter 5,...

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Unformatted text preview: 234 CHAPTER 5. IMPULSE AND MOMENTUM 5.6 Chapter 5, Problem 6 Problem: Two balls of mass m1 = 4m and m2 = 3m and coefficient of restitution e approach each other with velocities v1 = V i and v2 = − 4 V i. Determine their velocities after the impact, v1 and v2 , 3 in terms of V and e. Compute the velocities for a perfectly-plastic impact, a perfectly-elastic impact, and an impact with e = 1 . 2 ...... ..... .. ..... ................. .......................... ... . ................................. ................................... ....................................... 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 = − 4 V , while m1 = 4m and m2 = 3m, momentum conservation along the line of impact tells us 3 that 4 4mV − 3m V = 4mv1 + 3mv2 3 Dividing through by m yields 4v1 + 3v2 = 0 Impact Relation. For a coefficient of restitution equal to e, we have 7 eV 3 To complete the solution, we solve for v2 in terms of v1 from the impact relation, which yields v2 − v1 = e (v1 − v2 ) = 7 v2 = v1 + eV 3 Substituting for v2 in the momentum equation tells us that 7 4v1 + 3 v1 + eV 3 Again using the impact relation, we have 7 v2 = −eV + eV 3 Therefore, the velocity vectors after the impact are v1 =⇒ v2 = 4 eV 3 =0 =⇒ v1 = −eV = −eV i 4 v2 = eV i 3 For a perfectly-plastic impact, we know that e = 0, while e = 1 for a perfectly-elastic impact. Thus, we have the following velocities for e = 0, e = 1 and e = 1 . 2 v1 = 0 and v2 = 0, Perfectly-plastic impact Perfectly-elastic impact Impact with e = 1 2 4 V i, 3 1 2 v1 = − V i and v2 = V i, 2 3 v1 = − V i and v2 = ...
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This note was uploaded on 12/16/2010 for the course AME 301 at USC.

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