# If ball bouncing looks like a modulated absolute value of a sine wave, then why not model bouncing this way?

If ball bouncing looks like a modulated abs of sine wave, then why not model bouncing this way?

Rather than this way:

• It looks like a modulated sine wave, but it isn't! A graph of $x^3$ looks a lot like a graph of $x^5$, but there are important mathematical differences between them. In particular, the gradient of a sine wave is also sinusoidal (it's a cosine), but it's clear that here the velocity is not sinusoidal at all! Nov 26, 2017 at 19:51
• @gj255 What you mean it looks like but isn't? I mean, why can't it be replicated the modulation way? Nov 26, 2017 at 19:52
• mavavilj, in between the bounces, isn't the trajectory parabolic? Nov 26, 2017 at 19:57
• mavavilj, think about it; if the shape in between bounces were the first half of a sine cycle, the derivative would be the first half of a cosine cycle. But the velocity plot (time derivative of position) isn't remotely sinusoidal, it's a line. Also note that the time interval between bounces decreases with time and so, a modulated sinusoidal approximation to this must be of the form $|A(t)\cdot\sin(\omega(t)\,t)|$, i.e., both the modulation and amplitude change with time Nov 26, 2017 at 20:13
• Something to keep in mind when designing a simulation. Just writing the differential equations that approximate a systems behavior does not necessarily enforce other fundamental constraints, for example the conservation of energy. The result being that the simulation result is biased or unstable. This can be particularly amplified in nonlinear systems. One way to correct this issue is to use symplectic integration. Symplectic integration helps preserve energy and momentum conservation. Nov 27, 2017 at 15:45

Here is a plot of a sine curve overtop a parabolic curve:

Close, but clearly different. Also, if you wanted to show multiple bounces, you'd have to modulate not only the amplitude but also the frequency. This would become problematic because a continuously modulated frequency would lead to asymmetric bumps like this:

You could modulate the period and amplitude in a piecewise fashion so that they change for each bounce, but at that point your solution has become more complicated than the correct one.

• The point about the frequency changing for an actual bouncing ball is probably even more important than the point about the shape. Both are good points. Nov 26, 2017 at 21:40

Looks are deceiving. But to really answer it really depends on the intended purpose of your modeling.

If for example the purpose is for simulation in the cinema, your choice of a rectified sine wave might be 'real' enough. The audience believes it to be real, and everybody is happy.

But if you wanted a flight crew to reach Mars, you better stick with Newton and the real physics equations. Any other approximation could be deadly.