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This is the specific question I refer to (exam practice):

Particle P has mass 3kg and particle Q has mass 2kg. The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, P has speed 3 ms^–1 and Q has speed 2 ms^–1. Immediately after the collision, both particles move in the same direction and the difference in their speeds is 1 ms^–1.

I did the following to (correctly) calculate the speed of each particle:

3kg * 3ms^-1 + 2kg * -2ms^-1 = 3kg * v + 2kg * (v + 1)
v = velocity of particle P = 0.6ms^-1
v + 1 = velocity of particle Q = 1.6ms^-1

My question is this: how do I know that the greater speed (v + 1) is for particle Q? Is it because it had the greater momentum before the collision, so it's supposed to have the greater velocity after the collision? If I assume that particle P has the greater velocity after the collision, the answer is different (and incorrect).

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The physical answer is that P can't pass through Q, so if there is a difference Q must be faster. – Ross Millikan Feb 12 '13 at 0:11
@Ross, how do you know which direction they're going, after the collision? – Gerry Myerson Feb 12 '13 at 0:13
@GerryMyerson: The net momentum is positive, so they must be going that way. – Ross Millikan Feb 12 '13 at 0:15
@RossMillikan: Brilliant, I cannot believe I hadn't thought of that. Unless there is a "proper" way to determine the faster particle with the information given, then I'd take that as the answer. – Wk_of_Angmar Feb 12 '13 at 0:17
up vote 1 down vote accepted

You don't, but you can see what happens if you assume particle P has greater (more positive) velocity. Then you get that particle Q has velocity $.4\ \textrm{m/s}$ (notice that this is signed velocity, not magnitude), and particle P, $1.4\ \textrm{m/s}$. This is nonsense, though, (as Ross pointed out) since you assumed WLOG that $P$ started to the left of $Q$.

Generally speaking, for elastic collision of two particles the ending velocity is uniquely determined by conservation of momentum and conservation of energy. For inelastic collisions, such as the one here, you must use the information given in the problem to determine the final velocity. There is no universal principle about which particle must move faster than the other.

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both particles move in the same direction and the difference in their speeds is 1 ms^–1.

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