Coefficient of restitution bouncing ball I was researching for my physics investigatory project and I have very big confusion.
The ratio between the initial height of the ball (the independent variable) and the height the ball bounces (the dependent variable)
$$\mathrm{COR} = e =\sqrt{\dfrac{\text{final height}}{\text{initial height}}}.$$
As the drop height increases, COR should decrease or will it be equal, regardless of the changes in the size of the ball and the drop height?
Also if the drop height increases will the bounce height also increase?
 A: The COR is technically only defined for collisions. In this case the ball collides with the floor so we use the COR. The COR is more precisely defined as
$$e=\frac{v_f}{v_i}$$
where $v_i$ is the (relative) velocity before the collision and $v_f$ is the (relative) velocity after the collision. With relative I mean the relative velocity between the two objects. If one ball was moving with $v=5$ to the right and another ball with $v=1$ to the left, the relative velocity would be $5-(-1)=6$: the rate at which the gap between them is closing.
The COR tells you how much energy is lost during a collision. For $e=1$ no energy is lost and for $e=0$ all energy is lost. A collision between two pieces of clay which stick together after the collision is an example where all energy is lost. You can also write the COR using the kinetic energies using $v=\sqrt{2E_k/m}$, which gives
$$e=\sqrt{\frac{E_{k,final}}{E_{k,initial}}}$$
You can then use conservation of energy,
$$E=\tfrac 1 2mv^2+mgh,$$
to derive the equation you mentioned. You have to relate the energy before the drop to the energy just before the collision. This becomes
$$e=\sqrt{\frac{h_f}{h_i}}.$$
Note that this equation is only a valid expression for the COR if no energy is lost during the fall. Generally there will be air friction which can dissipate energy. If no energy is lost to friction, $e$ will be independent of drop height. Now my question for you is: if there is air friction, will the COR decrease or increase if you increase the drop height?
