# If the velocity of particle $A$ exceeds that of $B$, is the acceleration of $A$ greater than $B$?

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)

• 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). – wataya Aug 25 '13 at 6:15
• No. The question says that *at some point$A's velocity exceeds B's. It wasn't higher to begin with. – Gerard Aug 25 '13 at 6:24 • You should make that clear. The origin is also some point. – wataya Aug 25 '13 at 6:26 • Are you talking about average or momentary accelerations? – wataya Aug 25 '13 at 6:33 • Acceleration of both particles is assumed to be constant. If you are unclear about the question: This is question 13 from Chapter 2 Resnick Halliday Physics. – Gerard Aug 25 '13 at 6:46 ## 1 Answer No. It does not necessarily mean that the acceleration of$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.