(idea) by Sverre Mon Feb 11 2002 at 18:54:19
Since we don't make mistakes when we do physics, we daringly expand the problem to involve a collision between two entities with different masses (still with only one dimension).

Before:

o  --->  <-- o
mA  vA0   vB0 mB
After:
<-- o    o --->
 vA mA   mB vB

Now we get the following:
I) mA * vA + mB * vB = mA * vA0 + mB * vB0 (conservation of momentum)
II) 1/2 * mA * vA2 + 1/2 * mB * vB2 = 1/2 * mA * vA02 + 1/2 * mB * vB02 (phew, conservation of energy)
Note that we can drop the factor 1/2 everywhere by multiplying both sides by two.

Now for an example: You are rolling a bowling ball towards another, lying still. The two collide elastically. The ball you tossed is somewhat heavier than the other. In fact,
mA = 5 kg
vA0 = 30 kph
mB = 3 kg
vB0 = 0 kph

What happens next? 

I) vA = (mA * vA0 + mB * vB0 -  mB * vB)/mA)
      = (150 kg/kph - vB * 3 kg) / 5 kg
      = 30 kph - 3/5 * vB
... which may not come as a surprise.

Now insert this in (II), which we leave as an exercise for the clever student, and you get two solutions. Again, one solution is uninteresting - you missed, and the balls continue with their starting velocities. The other solution is (unless I'm completely wrong)

vB = 31.25 kph
vA = 11.25 kph

In general for this example, if the tossed ball is heavier, they both continue in the same direction. If the tossed ball is lighter, it bounces back, giving a smaller speed to the other ball.

I like it!