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Two particles $A, B$ are travelling along parallel straight paths. At some point, the velocity of $A$ exceeds that of $B$. Does this necessarily mean that the acceleration of $A$ is greater than the acceleration of $B$?

If you look at the $v - t$ graph of the two particles, the lines would intersect. Probably, starting off, the velocity of $B$ would be greater, but since the slope of the velocity of $A$ would be greater it would intersect with the graph of $B$ and exceed it. I couldn't think of any other situation. So, my conclusion was that the acceleration has to be greater. But my textbook says otherwise. How come?

EDIT: This is question 13 from chapter 2 in Resnick halliday physics.

To clarify: the problem does NOT assume that initally A's velocity was lower than B's. (See comments)

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    $\begingroup$ Not sure if I understand your question correctly. If A starts with a high velocity and travels at constant speed and B starts from rest and is accelerated, than obviously A's velocity is greater than that of B and B's acceleration (>0) exceeds that of A (=0). $\endgroup$ – wataya Aug 25 '13 at 6:15
  • $\begingroup$ No. The question says that *at some point$ A's velocity exceeds B's. It wasn't higher to begin with. $\endgroup$ – Gerard Aug 25 '13 at 6:24
  • $\begingroup$ You should make that clear. The origin is also some point. $\endgroup$ – wataya Aug 25 '13 at 6:26
  • $\begingroup$ Are you talking about average or momentary accelerations? $\endgroup$ – wataya Aug 25 '13 at 6:33
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    $\begingroup$ Acceleration of both particles is assumed to be constant. If you are unclear about the question: This is question 13 from Chapter 2 Resnick Halliday Physics. $\endgroup$ – Gerard Aug 25 '13 at 6:46
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No. It does not necessarily mean that the acceleration of $A$ is greater than the acceleration of $B$. Here's an explicit counterexample:

Object $A$ is moving at $10\,\mathrm{m/s}$ with constant velocity while object $B$ is moving at $5\,\mathrm{m/s}$ with an acceleration of $1\, \mathrm m/\mathrm s^2$. In this case, the acceleration of $A$ is zero, so $B$'s acceleration is greater, but it's velocity is lower.

Note that the initial conditions of the motions of the two objects are irrelevant; we're talking about instantaneous velocities and accelerations, and given any two objects, one can completely independently pick their velocities and accelerations.

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