What does it mean empirically if a bouncing ball's frequency depends on height?

That the trajectory's curvature will vary based on height?

I.e. we have a bouncing ball equation, where the ball's velocity

$u_t=v$, $v$ velocity

$v_t =$ function of height, e.g. $-g-h^2$, where $g$ is gravity.

The frequency is found by forming the Jacobian and solving the system's eigenvalues.

Since $h$ is retained in the Jacobian, then it's in the eigenvalues as well.

So we have eigenvalues:

$\lambda = \pm i \sqrt{2} \sqrt{h}=\pm i\omega$

and since $\omega$ is the frequency, then now it depends on $h$.

So the higher the $h$ the higher the frequency? But what kind of empirical sense does this make?

  • $\begingroup$ it means the force pulling the ball back to the ground is not proportional to its height. can't imagine a situation where it would be. $\endgroup$ – Ben51 Feb 15 at 15:53
  • $\begingroup$ Balls on springs. There you go $\endgroup$ – shai horowitz Feb 17 at 9:05

The frequency of bounce depends on the time required for the ball to return to the ground after each bounce. If it is bouncing higher, it takes the ball longer to go up and then come back down.


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