A ball dropped from 1.8 m onto a floor bounces back up to 1.15 m.

When only conservative forces (gravity, springs) do work, mechanical energy is conserved: $K_1 + U_1 = K_2 + U_2$. Forces that do no work, like a smooth surface's push or a string at right angles to the motion, do not spoil it.

The coefficient of restitution is $e = \dfrac{\text{speed of separation}}{\text{speed of approach}}$. A ball bouncing on a floor keeps the fraction $e$ of its speed and $e^2$ of its height each bounce, so $e = \sqrt{\dfrac{h_1}{h_0}}$.