Real world examples for projectile thrown upwards or downwards 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 
$$
s(t) = \frac{1}{2} g t^2 + v_0 t
$$
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... 
 A: Another one:

A magician is locked in a wooden chest and is fired straight up from a very powerful cannon. He has a parachute to land safely, but it won't work if it's opened less than 25 meters from the ground. The magician tested the cannon (with a crash dummy, of course), and it can launch him up to 50 meters high. How much time will he have to escape from the chest?

A different version, less contrived but a little more difficult:

...The magician tested the cannon (with a crash dummy, of course), and it takes the dummy 6 seconds to crash bask into the ground. How much time will he have to escape from the chest?

A: If the initial velocity $v_0$ is allowed to be zero, here is one: 

You are standing next to a very deep hole, and wonder what is the depth $h$? You drop a stone into the hole, and hear a delayed sound (of the stone hitting the bottom) after time $T$ on your stopwatch. Given the gravitational constant $g$, given the speed of sound $c$, and by ignoring air-resistance on the stone, calculate the depth $h$.

Solution: 
$$ h ~=~ cT + \frac{c^2}{g}\left[ 1- \sqrt{1+ \frac{2gT}{c}}\right].$$
A: Not related to guns or projectiles I have recently used the 2D version of this equation to make sure oil is squirted into an engine at the correct location using an "oil jet". If the oil did not land in the correct location, then the engine components will overheat and eventually fail (bad). So in a design sense, the equation allowed me to find the correct position for the oil squirters. The oil comes out of the orifice in a steady stream, with known exit velocity based on the oil pressure behind the hole. 
The reason velocity has to be known is because of Newtons Laws of motion which help us take an frozen instant in time and calculate accelerations from positions and velocities. In its core, determinism means if I know the state now, I can predict it in the near future. This is all it is done with the projectile equations.
A: This one could work (and it's based on real events):

You are at a friend's apartment and you are about to leave. He's very lazy, so he gives you the keys of the door at the ground floor. You get out of the building and you try to toss the keys back to him through an open window (or at his balcony), that's about 9 meters high. How fast should you throw the keys?

Or, if a small bending of the rules is allowed:

...that's about 9 meters high. You've played throwing rocks before, so you know that you can't toss objects similar to the keys further than about 30 meters. Can you throw that high? What's the highest floor that you could throw the keys at?

Of course, check that the numbers work for you.
A: Here's one example that almost fits your qualifications: one time on Mythbusters they investigated the behavior of bullets fired straight up in the air, or what people would consider "straight up" which in practice means at a small angle away from truly vertical. The main thrust of the investigation was to determine how far away a bullet would travel horizontally, in other words to determine the size of the "danger zone" in which bystanders are liable to be hit by the falling bullet. And of course, to do that you need to calculate the flight time using the muzzle velocity of the bullet. It's probably safe to assume some small angle like $3^\circ$.
I forgot to mention at first: this wouldn't be a realistic calculation because air resistance is very significant in reality. But you can still use it as a kinematic example.
A: A jumping astronaut is an example that might be interesting. I always found it very exciting to watch documentaries about the Apollo missions and watch how the astronauts are jumping around on the moon. From our perspective it looks almost cheerful, as they can jump higher and the whole movement looks therefore much slower. Considering that the inertia is the same whether your are on the moon or not it seems not far fetched that the initial velocity from the astronauts does not really depend much on gravity and therefore the maximum height and time that the astronaut needs to complete the jump is mainly determined by the moon's lower gravity. So for an example the students could perform small jumps in the classroom and measure the time until they hit the ground again (with a backpack to create similar conditions). Now compare this time with some videos from the Apollo missions and try to determine the change in $g$ by comparing the different times. While this will not be precise I would guess you should end up in the right ballpark of $$g_{moon} = 0.167 \cdot g_{earth} $$
A: Since the previous example was biased, hence i am adding another example,
Suppose there is a hot air baloon at a point a,b in 2d world.
and a gun which is kept at origin is designed to hit the baloon.
then find the minimum velocity of the bullet of gun,(which follows a projectile path) so that it hits the hot air baloon.
{This is problem of inclined plane projectile,analogy} 
A: (a bit like the oil jet answer..) What about a fountain which directs a jet of water essentially straight up?
Pros: an everyday real-world non-contrived example
Cons:
- it involves a stream rather than a particle.  The math of following a conceptual "fluid particle" is the same but it would probably be more satisfying to have an example that involves following a point mass


*

*you wouldn't directly know the speed of the jet at the base of the fountain but it would be obviously calculable (fluid mechanics, pressure drop is 0.5 * rho * v^2).

