I like to spend time outside when possible. On a recent adventure I took a couple of the kids to check out some trails near my mother's house. This particular place was pretty nice. It had a lake with some cliffs you could walk along. Note: Do not jump off the cliffs into the water—there is a $500 fine for that (at least that's what the sign said).
As we were standing near the edge of one of these cliffs, my daughter said that it wouldn't be too bad to jump off—it's not that high. I was pretty sure it was higher than she thought it is. But I don't have to just guess; we can measure the height with just a rock, my phone, and physics.
Here's what you do. Take your phone and get ready to record some video. Now drop the rock from rest so that it falls into the water. If you have to toss the rock, that's fine, as long as you only throw it horizontally. Don't throw it up or down—this will give an inaccurate measurement for the height. The only thing you need from the video is the time it takes the rock to fall and hit the water. From this time, we can calculate the height.
For my cliff, I got a free fall time of 1.3 seconds (I got this using Tracker Video Analysis—but there are lots of other programs to get the time).
After you let go of the rock, there is essentially only one force acting on it—the gravitational force. This is a force that depends on the mass of the rock and the gravitational field (with a value on the surface of the Earth of 9.8 Newton/kilogram). We usually use the symbol g to represent this value. Since there is only one force on the rock, the rock will continue to speed up (accelerate)—that's what constant forces do to an object. The acceleration of an object depends on both the force AND the mass. Since both the force and the acceleration depend on mass, in this case it cancels and you get an acceleration of g m/s2 in the vertical direction. That's why different mass rocks would hit the water at the same time (assuming the air resistance is negligible).
Now we know the acceleration of the falling rock. The acceleration describes the rate that the velocity changes. In this case, the starting velocity is zero m/s and the final velocity is unknown. This means that I can use the acceleration to find an expression for the final velocity (even though I don't really care about this). Oh, I am going to call the downward direction the positive direction just for fun and because coordinate systems aren't real.
If it makes you happy, you could put in the value for your rock time and g—but you don't have to right now. Instead, I am going to use the definition of the average velocity (in one dimension). Actually, there are two definitions of the average velocity. There is the rate of change of position and there is the actual average.
You could stop here if you already calculated the final rock velocity—but I didn't do that. Instead, I can put my expression for this velocity into the this equation for the height and I get.
Boom. This is my answer. I will put my value of 1.3 seconds for the time and 9.8 for g and I get a cliff height of 8.28 meters (or 27.2 feet). That's pretty high—I'm sure it's higher than my daughter would have guessed.
But wait! Why didn't I just make this whole thing shorter and use the kinematic equations to find the height? Well, that's what I did. However, I didn't just use the final equation I also showed you where that equation comes from. See—accidental physics.
Note: If you take a tiny rock and drop it, this might not work. The mass to surface area ratio for super tiny rocks means that the air resistance will be a significant force and the rock won't fall with a constant acceleration. Instead, take a good sized rock to drop. Oh, make sure no one is below you—getting hit by a rock is bad.
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