Calculating height using energy

- [Instructor] So I have an uncompressed spring here and the spring has a spring constant of four Newtons per meter. Then I take a 10 gram mass, a 10 gram ball, and I put it at the top of the spring and I push down to compress that spring by 10 centimeters. And so let's call that scenario one right over there, where our mass is on top of this compressed spring. And then let's say, we let go. And then the spring launches this mass into the air, and then this mass is gonna hit some maximum height. And let's call that scenario two when we are hitting that maximum height. And my question to you is, what is that maximum height based on all of the information that I have given you here? And 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 gonna 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 gonna be conserved, but what's that total energy going to be made up of? Well, it's gonna have some potential energy and it's going to have some kinetic energy. And so 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. And there might be other energies here that 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. There's gravitational potential energy, and there is elastic potential energy due to the fact that this mass is sitting on a compressed spring. So our gravitational potential energy is going to be our mass times the strength of our gravitational field, times our height in position one, and our elastic potential energy is going to be equal to one half times our spring constant, times how much that spring is compressed, squared. And so this is all of our potential energy in scenario one, right over here. And then we add our kinetic energy. So plus, we have one half times our mass, times our velocity in scenario one, squared. And so the sum of this is gonna be the sum of all of this in scenario two. And so 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's spring is compressed in scenario two squared, plus one half times our mass, times our velocity in scenario two, squared. And just as a reminder, what we have right over here, this 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, H one as being equal to zero, which will simplify this dramatically. Cause if H one is zero, then this term right over here, zero, we also know that our velocity is zero in our starting scenario. So that would make our kinetic energy zero. So the left-hand side is really all about our elastic potential energy. So it's gonna be one half times our spring constant, times how much we have compressed the spring in scenario one, squared. And then 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. And 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 that's gonna be zero. And so you have a scenario where our initial elastic potential energy is going to be equal to our scenario two, gravitational potential energy. And so 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 1/2 k delta x, one squared, all of that over mg. And 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 four 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. So I wanna convert this 10 centimeters to meters. Well, that's gonna 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 so that is gonna be squared. And then all of that over, what is our mass? And 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. And when you calculate it, this is approximately 0.2 meters. And the units actually all do work out. Because if you look at Newtons as being the same as kilogram meters per second squared, this kilograms cancels with that kilograms right over there. You can see that this per second squared is gonna cancel with that per second squared right over there. And then this meter is going to cancel out with that meter. And so 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.