Calculating height using energy | Modeling Energy | High School Physics | Khan Academy

So I have an uncompressed spring here, and this spring has a spring constant of 4 newtons per meter. Then, I take a 10 gram mass, a 10 gram ball, and I put it at the top of the spring. I push down to compress that spring by 10 centimeters. Let's call that scenario one, right over there, where our mass is on top of this compressed spring.

Then, let's say we let go, and the spring launches this mass into the air. This mass is going to hit some maximum height, and let's call that scenario two when we are hitting that maximum height. My question to you is: what is that maximum height based on all of the information that I have given you here?

I'll give you a hint: what we have to think about is the idea that energy is conserved. The total energy in scenario one is going to be equal to the total energy in scenario two. So pause this video and see if you can figure out what that maximum height is going to be.

All right, now let's work through this together. I told you that the total energy is going to be conserved, but what's that total energy going to be made up of? Well, it's going to have some potential energy and it's going to have some kinetic energy.

Another way to think about it is our potential energy in scenario one plus our kinetic energy in scenario one needs to be equal to our potential energy in scenario two plus our kinetic energy in scenario two. There might be other energies here; you could think about heat due to friction with the air, but we're not going to. We're going to ignore those to just simplify the problem. We can assume that this is happening in a vacuum; that might help us a little bit.

So if we think about the potential energies, there's actually two types of potential energies at play: gravitational potential energy and elastic potential energy due to the fact that this mass is sitting on a compressed spring. Our gravitational potential energy is going to be our mass times the strength of our gravitational field times our height in position one. Our elastic potential energy is going to be equal to one-half times our spring constant times how much that spring is compressed squared.

This is all of our potential energy in scenario one right over here. Then we add our kinetic energy, so plus we have one-half times our mass times our velocity in scenario one squared. The sum of this is going to be the sum of all of this in scenario two.

That is going to be equal to mass times g times height in scenario two plus one-half times our spring constant times how much that spring is compressed in scenario two squared plus one-half times our mass times our velocity in scenario two squared. Just as a reminder, what we have right over here is our potential energy in scenario two, and then our kinetic energy is right over there.

Now, this might look really hairy and daunting, but there's a lot of simplification here. We can define our starting point right over here, h1, as being equal to zero, which will simplify this dramatically because if h1 is 0 then this term right over here is 0. We also know that our velocity is 0 in our starting scenario, so that would make our kinetic energy zero.

The left-hand side is really all about our elastic potential energy. It's going to be one-half times our spring constant times how much we have compressed the spring in scenario one squared. On the right-hand side, what's going on? Well, in this scenario, our spring is no longer compressed, so our elastic potential energy is now zero.

What about our kinetic energy? Well, at maximum height right over here, your ball is actually stationary for an instant, for a moment. It's right at that moment where it is going from moving up to starting to move down, and so our velocity is actually zero right over here. So v2 is actually zero just like v1 was zero.

So, you have a scenario where our initial elastic potential energy is going to be equal to our scenario two gravitational potential energy. We just need to solve for this right over here; that is going to be our maximum height. To do that, we can divide both sides by mg, and we get h2 is equal to one-half k delta x1 squared over mg.

We know what all of these things are, and I'll write it out with the units. This is going to be equal to one-half times our spring constant, which is 4 newtons per meter. Now our change in x is 10 centimeters, but we have to be very careful. We can't just put a 10 centimeters here and then square it; we want the units to match up.

So we're not dealing with centimeters and grams; we're dealing with meters and kilograms. I want to convert this 10 centimeters to meters. Well, that's going to be 0.1 meters, so I will write this as times 0.1. That's how much the spring is compressed; that's our delta x in terms of meters, and that is going to be squared.

Now, all of that is over what is our mass? Once again, we want to express our mass in terms of kilograms, so our mass is 0.01 kilograms. g is 9.8 meters per second squared.

When you calculate it, this is approximately 0.2 meters. The units actually all do work out because if you look at newtons as being the same as kilograms meters per second squared, this kilograms cancels with that kilograms right over there. You can see that this per second squared is going to cancel with that per second squared right over there, and then this meter is going to cancel out with that meter.

What you're left with is a meter squared divided by meters, which is just going to leave you with meters, just like that. And we're done. We figured out the maximum height using just our knowledge of the conservation of energy is approximately 0.2 meters.