A bouncing ball

A ball of mass $m$ is thrown and bounces on the floor. The forces acting on this ball are gravity $-mg\hat{z}$, a drag force proportional to the velocity$-a\vec{v}$, and the force exerted by the floor when it bounces. The floor can be described as an elastic material that pushes up with a force $-kzH(-z)\hat{z}$. Here $k$ is the elastic contant and the Heaviside function $H(-z)$ ensures that this force only if the ball falls below $z=0$.

The larger the elastic constant $k$ is, the less the ball will go below $z=0$ and the simulation will approach bounces off a hard floor. However, some care must be taken in choosing the time step $\Delta t$. It should be much smaller than $2\pi\sqrt{m/k}$ so that the solver can describe the bounces off the floor properly. Some differential equation solvers adjust the time step automatically to account for problems like this.

$\vec{F} = m\frac{d^2\vec{r}}{dt^2} = -a\vec{v}-mg\hat{z} -kzH(-z)\hat{z}$