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# “Speed difference” conservation

Instead of saying that elastic collisions ‘“conserve” the speeds’ it is more correct to say that they ‘“conserve” the speed difference’ between the two objects. Also, it turns out that this difference is “flipped” during an impact, since the objects now are influenced to go back to where they came from. We can write that as (see proof 31):

\begin{align} \Delta v_\text{before}&=-\Delta v_\text{after}\quad\Leftrightarrow\\ v_\text{A,before}- v_\text{B,before}&= -(v_\text{A,after}- v_\text{B,after})\end{align}

Let’s always remember that this “speed conservation” equation only holds true for elastic collisions.

• Two billiard balls that are equal will directly “switch” speeds when they collide (a ball hitting another stationary ball will now be stationary while the other ball moves off).
• But, naturally a billiard ball hitting something very heavy, like a bowling ball, will not make that bowling ball move backwards. It will merely cause a slight change in the bowling ball’s speed and then itself move backwards at a slightly lower speed.

Impacting a wall is a special case of this: If we think of a wall as an “incredibly heavy” object, then a ball hitting it will come flying back at the same speed it impacted with – the wall itself will be too heavy to gain any noticeable speed.1

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