# Find the height of a bouncing ball at any given time

1. Jun 9, 2014

### Tizyo

1. The problem statement, all variables and given/known data

Find an analytical model for the height of a bouncing ball to be able to find the the height of the ball at any given time. In other words, as I understand, write an equation that would define the trajectory of a bouncing ball.
Let's say the coefficient of restitution (COR) is 0.9 (it's a table tennis ball)

2. Relevant equations

3. The attempt at a solution

Im kinda stuck on this, this is a bit beyond me, but I thought maybe somehow transforming the absolute value of y=cos(x); or using a bunch of quadratic equations to represent each bounce, but that doesn't seem right.

2. Jun 9, 2014

### haruspex

Start with just one bounce (i.e. from leaving the floor to returning to it). Please post the equation for that.
When you say 'analytical model', what form of answer do you think is required?

3. Jun 9, 2014

### Tizyo

Analytical model - developing an equation(s) to fit the data that I collected. I threw a ping pong ball while filming it, analysed the video using LoggerPro, i.e. I plotted the position of the ball in each frame of the video. And now I have to find a function (first on paper, and then test with technology) that would fit the points that I got.

The more I think about it, the more it looks like something like y=abs((2/x)*sin(x)) is similar to the bouncing ball trajectory, expect the distance between bounces doesn't change and the height diminishes too fast. And I don't know what variables/coefficients(numbers in the equation) should the it have.

I don't really know how to make the equation for one bounce... could you please explain or show it to me? I am kinda lost :/

4. Jun 9, 2014

### Staff: Mentor

I would start with dropping the ball and it hitting the ground. Is this supposed to be a 2-D problem, or a 1-D problem?

If a 2-D problem, you can probably just assume a constant horizontal velocity Vx. What is the equation for Vy(t) from the drop point (where V=0 at t=0) down to when it hits the floor? What is the final velocity just as it hits the floor?

Then what will it's initial velocity Vy be as it bounces up? Use that as your initial velocity for the next parabolic path, and write that equation until the next contact with the floor. Then what will the velocity be as it starts its next trip up? And so on....

5. Jun 9, 2014

### haruspex

I'm not sure what your 'x' is. It only asks for height as a function of time, y = f(t). Are you using x for time?
I'm very surprised you don't seem to know the standard SUVAT equations. Each trajectory is the result of a constant acceleration, g. (Hint: nothing to do with trig functions.) It's very easy to write down the equation for one trajectory, given the conditions just after it bounced; the problem comes in finding a function to represent the height at any time t from the initial drop.
No 'analytic' function is going to give you that. It will have to involve something more algorithmic in nature. Specifically, you need a way to convert a time to a number of impacts (a whole number) and the time since last impact. There are 'mathematical' functions, like floor and ceil, which can handle this, but I would not call them analytic.

6. Jun 10, 2014

### Ray Vickson

If the ball is dropped from rest from an initial height of $h_0$ it hits the floor first at time $t_1$, then bounces up to some height $h_1$ and then falls to next hit the floor at some time $t_2$, etc. So, your final equation will have a form like
$$h(t) = \left\{ \begin{array}{rl} f_1(t), & 0 < t \leq t_1 \\ f_2(t), & t_1 < t \leq t_2 \\ \vdots \end{array} \right.$$
Here the functions $f_1(t), f_2(t), \ldots$ need to be supplied by you, and you also need to figure out the bounce times $t_1, t_2, \ldots$.

After you have figured out the expression as shown above, then you can worry about whether it can be expressed in some type of one-line formula or something similar. Maybe you can use so-called Heaviside functions to re-write everything, but that should not be your first concern. Your first concern should be to understand the problem and figure out the functions $f_i(t)$ and hitting times $t_i$. Do not just guess; work these out from first principles, for example, using some elementary physics developments about motion in a constant gravitational field---assuming no air friction (which would make the problem a lot harder).

Last edited: Jun 10, 2014