Source: High School Physics Olympiad
Problem Sets:
Problem
Several small balls are thrown simultaneously from the same point in air, in all directions, each with the same speed $v_0$. Air resistance is neglected.
All balls lie on a sphere of radius $v_0 t$ centered on a point in free fall from the launch point
Take the common launch point as the origin. A ball thrown with initial velocity $\vec{v}_{0i}$ (with $|\vec{v}_{0i}| = v_0$) has position:
$$\vec{r}_i = \vec{v}_{0i}t + \frac{1}{2}\vec{g}t^2$$Define the point $\vec{r}_c = \frac{1}{2}\vec{g}t^2$, which is a point falling freely from rest from the launch point. Then for every ball:
$$\vec{r}_i - \vec{r}_c = \vec{v}_{0i}t \quad\Rightarrow\quad |\vec{r}_i - \vec{r}_c| = v_0 t$$Since this distance is the same for all balls, at any time $t$ they all lie on a sphere of radius $v_0 t$ centered at $\vec{r}_c$, and the center $\vec{r}_c$ undergoes free fall. $\blacksquare$