# Closed form of the position of a bouncing ball

Homework Statement:
This is not a physics homework per se, but I'm implementing an explosion shader in OpenGL for class and I want the triangles to bounce when hitting the y=0 plane. In my current setup it is not possible to save the triangle's velocity or position and so each frame I compute the position of each triangle with a ballistic equation, and for now I simply set y=0 for triangles that would go below the plane y=0.

Is there a closed form for the position (or rather height as the x axis is irrelevant here) of a bouncing ball, and if not why can't there be one?
Relevant Equations:
Ballistic equation: ##y = g * t * t + v_0 * t + y_0##
Elasticity of ball: ##e##
Velocity after bounce: ##v_{after} = e * -v_{before}##
I know that the height before the first bounce will be ##y = g * t * t + v_0 * t + y_0##.
After the first bounce, I can find y by pretending the ball was thrown from the ground with velocity ##e * -v_f## with ##v_f## being the velocity of the ball when hitting the ground, but I have to reset the origin of time by subtracting the time it took until the first bounce (##t_1##) so ##y = g * (t - t_1) * (t - t_1) + (e * -v_f) * (t - t_1)##. I can repeat this for as many bounces as needed so this is easy to do in a step-by-step simulation, but I can't seem to figure out how to find a rigorous closed form from here.

haruspex
Homework Helper
Gold Member
2020 Award
how to find a rigorous closed form
Can you figure out the fraction of KE lost each bounce?

Can you figure out the fraction of KE lost each bounce?
Right before bouncing, KE is ##\frac{1}{2}mv_{before}^2## and right after the bounce it is ##\frac{1}{2}e^2mv_{before}^2##, thus the fraction of KE lost each bounce should be ##1-e^2##, correct?

haruspex