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.

 

[haruspex]

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?

 

[haruspex]

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?

Right.
Next, can you figure out the duration of the nth bounce?