# Relation between velocity magnitude and the pitch of the collision sound

I'm doing a simulation involving collisions, I'm using glass marbles, based on testing I've noticed that the higher the magnitude of the velocity of the marble hitting another marble at rest the higher the pitch of the collision.

With a small magnitude, let's say you move one marble carefully to touch another marble the sound will be lower as well as the pitch of the sound, all these tests are based on trial and error, there are no math or physics equations involved, my question is, is my claim true? Do any of you know the proper relation between these 2 variables?

Neat question!

First some setup. The collision of marbles is within the domain of contact mechanics. When two spheres collide their surfaces both deform. This forms a circular contact area between them, with

$$A \propto F_n$$

The normal force $F_n$ being dependent on the velocity of the collision. Relying on the geometry of spheres we can also express the contact time $T$ of the two marbles as a function impact velocity, $v_0$:

$$T = k \frac{A}{r \ v_0}$$

In this case $r$ is the radius of the marble, and $k$ is a constant that depends on material properties.

In the end: shorter contact time $\rightarrow$ higher frequency $\rightarrow$ higher pitch.

One additional note. The sound of the collision is not coming from the resonant modes of the marbles. Their natural frequency of vibration too high to be audible to us. (You can read more about that here: https://physics.stackexchange.com/q/376988)

If you want to read more, here's a great slideshow involving the sound of colliding balls, but in a slightly different scenario. A classic paper on this topic is Richards 1979 On the prediction of impact noise, I: Acceleration noise.