# Closed form of the position of a bouncing ball

Bibibis
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.

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how to find a rigorous closed form
Can you figure out the fraction of KE lost each bounce?

Bibibis
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?