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Here's a bit of a fun physics paradox, which I will pose, and then answer below.

(These ideas were inspired by Grant Sanderson's fascinating video on how the digits of $\pi$ are hidden within elastic collisions.)

An object is travelling to the right, encounters an immovable barrier, undergoes a perfectly elastic collision, and returns to the left at an equal but opposite velocity.

Since the collision is perfectly elastic, kinetic energy and momentum are both conserved.

But the momentum after the collision is not equal to the momentum after the collision, because $p

e -p$. The net change, $\Delta p$ should be zero, but instead:

$$\Delta p = p_f - p_i = -p - p = -2p

e 0$$

What's up with that?