Consider an elastic collision.

Since the total KE remains the same after an elastic collision, we can write

Rearranging, we get

Elastic or not, total momentum is always conserved in any collision. So we can write

Dividing (2) by (3), we get

Notice that *u*_{1} – *u*_{2} represents the relative speed at which *m*_{1} approaches *m*_{2} before the collision, while *v*_{2} – *v*_{1} represents the relative speed at which *m*_{2} separates from *m*_{1} after the collision. So, in showing that equation (1) can be simplified into equation (4), we have stumbled upon a curious fact about elastic collisions: From the perspective of *m*_{2}, *m*_{1} came at it at the same speed as it went away.

I call equation (4) the RSOA=RSOS equation. (Relative speed of approach = relative speed of separation). Obviously, it is much simpler than equation (1). For this reason, do yourself a favour. Use the RSOA=RSOS equation instead of equation (1) to solve elastic collisions.

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**Video Explanation**

RSOA=RSOS

**Concept Test**

0463

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