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>> Let's make up a complicated problem. Okay, just out of the blue let's try the following. Let's include a spring in the problem, let's include gravity, and let's also include motion. Okay, and let's make up a problem. Let's say the problem looks like this. We've got a frictionless table that has a spring on it, and that spring is compressed at distance x from its equilibrium. And, then we're going to launch this thing off the table and it's going to go to the ground, and we want to figure out what is Vf for that box when it hits the ground. And, we will tell you that this is height h, and the box is a mass m. Alright, and let's say that this is frictionless. There's no friction on the table here. Alright. How do we do it? I'll give you a hint. It's the title of this chapter that we're working on. >> (student speaking) Only the height matters, so. >> Okay. >> (student speaking) Really. >> Andrew seems to think that only the height matters here. I'm not sure I totally agree with that, right? Because, we've compressed the spring in this for a dart gun configuration. It seems like that's going to matter too. Would you agree? >> (student speaking) No. It seems like most of your velocity is going to be from your height. >> Okay. Well, let's try it and find out. What sort of principle do we want to use here? I'll give you a hint. It's conservation of? >> (student speaking) Energy. >> Energy. Conservation of energy, right. And, again all this says is, whatever energy we have initially, has to be equal to whatever energy we have finally. Finally, meaning just before it hits the ground. So, what sort of energy do we have initially? Well, Andrew said that we have potential energy. Absolutely. Is there any other energy that we have initially? Yes, because we've compressed the spring, and so we have to include that right here. And, now we're going to launch it. So, this is before launch, and now we launch it and it falls to the ground. This is going to be our final position here. Final position is all kinetic energy. It's down at height zero, spring is gone, and so this is all we have on the right side of our equation. And, now we can solve this for Vf. Alright, let's rewrite it. One-half mVf squared equals mgh plus one-- half kx squared. I can, multiply everything by two. I can divide by m. And, I get Vf is equal to two gh plus.

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