I am preparing a physics course for high school about projectile motions. If a projectile moves with initial velocity $v_0$ in the gravitational field of the earth, the equation
$$ s(t) = 1/2 g t^2 + v_0 t $$
holds, where $s(t)$ is the travelled distance at time $t$. Now I am looking for real world applications of this equation (or the corresponding equation for the velocity).
More specifically I don't want any problems where the equation is somehow artificially embedded into a real world situation, for example
"you fire the projectile of a signal pistol with initial velocity $v_0=...$, at which height is the projectile at time $t=...$, what is the maximal height..." projectile signal pistol In that example it is not clear why you know the initial velocity or why you want to calculate the maximal height (indeed, I think in most cases the manufacturer writes the maximal height into the manual, but then why should you be interested in calculating the initial velocity?)
So the point is, in the problems/examples I am looking for, it should be clear, why one has the input data and why one wants to calculate other things using the equation of motion.
Please stick to one problem/example per answer. Further references for the context of the example would be nice.
Edit: I should make clear, that I want examples where $v_0 \neq 0$. I am only intersted in the upward - downward -case.
One example I was thinking of, was that of a vulcano that ejects stones. However I don't know much about vulcanos, so I don't know which initial data are known, and what people want so calculate...