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Cart A is moving with an initial velocity +v (in the positive direction) toward cart B, initially at rest. Both carts have equal mass and are on a frictionless surface. Which of the following statements correctly characterizes the velocity of the center of mass of the system before and after the collision?
  1. $\dfrac{+v}{2}$ before, $\dfrac{-v}{2}$, after
  2. $\dfrac{+v}{2}$ before, 0 after
  3. $\dfrac{+v}{2}$ before, $\dfrac{+v}{2}$ after
  4. 0 before, 0 after
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OpenStax College Physics for AP® Courses Solution, Chapter 8, Problem 19 (Test Prep for AP® Courses) (1:12)

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This is College Physics Answers with Shaun Dychko. We want to know how the center of mass in the system is moving before and after the collision of these two carts. This second cart is initially at rest and the first cart is moving towards it with a velocity of positive <i>v</i>. The center of mass is midway between them because they have equal mass and so the center of mass is right in the middle. So, the center of mass has to cover half the distance that the cart does in the same time, and so its velocity must be half that of the cart. So that narrows things down to options A, B, or C. Now, because there is no net external force on the system it means that the velocity of the center of mass can't change. There won't be any acceleration to it and so it's velocity after the collision will be the same as the velocity it had to begin with. So that means afterwards it must also be positive<i>v</i> over two. So the answer is C. The forces that the carts apply on each other are forces that are internal to the system so they don't have an effect on the center of mass.