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Push-Away Problems with Energy

Patrick Ford
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Hey guys, on the last couple videos we saw how to use momentum conservation to solve these kinds of push away problems. We have objects that are pushing away from each other. But in some of these problems you're actually gonna be asked for energy before or after a push away events. For example, this example we're gonna work down here has these two boxes that are pushed up against this spring. And in part B we want to calculate how much energy is stored inside of that spring. So I'm gonna show you really quickly have to deal with these push away problems with energy. And the main idea here is they're actually going to use to conservation equations. We're going to have to use momentum. We're always going to use momentum conservation with these problems, but we're also going to have to go back and use energy conservation as well. Let's go ahead and take a look at our example here, we've got these two blocks, three and four kg and their push up against this light spring and you release it in the spring basically is going to fire both of them off in opposite directions. So the blocks are gonna get released. So we want to draw a diagram for before and after. This is the before and afterwards. What happens is you sort of have these boxes that start pushing away from each other, like this, this four kg box goes like this and the three kg box is now going to the left. We actually know what the final velocity of this three kg block is, it's launched at 10 m per second. So what happens is I'm gonna call this M one and M two, which means that the one final is equal to negative 10 because it's going to the right, the left and then V two final here is actually what we want to solve for in part a the recoil speed of the four kg block. So this is where you look for in part A. And basically we're just gonna use momentum conservation. We know how to solve these kinds of push away problems. So once we ever before and after we're just going to write our momentum conservation. So for part A We're going to do M1 V1 initial plus M two V two initial equals M one V one final plus M two V two final. Now remember what happened to these kinds of problems? Is that initially the system is at rest. So right after you release your hand, the box, the blocks are still actually at rest, which means that the the initial velocity is equal to zero. What that means is that basically both of our left terms are actually going to go away and we have zero initial momentum. So the system's initial momentum is equal to zero. And that means that our momentum conservation equation is going to simplify because we know that the final momentum has to be zero as well. So basically what happens is we're gonna have negative M1 V1 final equals M two V two final. If the right block gains 10 momentum this way, the left bloc has to gain negative 10 momentum that way. And so they have to cancel each other. So we can we can basically solve for this V two final here and we can start plugging in values. So I have negative and what I have here is the mass is three and the velocity is 10. But is it 10 or is it negative 10? Remember we have to input this negative sign when we plug this in because it's important to keep track of the science. So we have three times negative 10 divided by four, which you end up getting here is that V two final is equal to 7.5 m per second. So notice how we got a positive number. And that should make some sense if we actually got a negative number. So if we forgot one of these minus signs, we would have gotten a negative number and that would have meant that this block is actually going to the left and that doesn't make any sense. So it's always a good idea to keep track of your numbers and make sure that makes sense. So this is a positive answer. Positive 7.5. And that's just using momentum conservation. Let's take a look at Part B. Now, part B is asking just to figure out how much energy is stored inside the spring. So what's going on here is that you've had these blocks that are pushed up against this spring and there's some elastic potential energy that is stored inside of the spring because they're compressed once you release. What happens is that these two blocks start firing away from each other, right? They separate like this and the spring decompress is which means that the final velocity you start, the final potential energy is equal to zero. So what happens is there's some stored energy here and then this spring is no longer storing the energy. Where does that energy go? Well, basically what happens is that you have these two blocks that have now mass and speed and therefore there's some kinetic energy. This is K two final and K one final. So basically we're going to use energy conservation to figure out the energy instead. So you're going to use K. Initial plus you initial plus work done by non conservative K final plus you final. Now what we said here is that the initial velocity of the system is zero. So there's no kinetic energy. The initial energy that's stored here is going to be elastic potential energy. There's no gravitational because everything is sort of on the horizontal plane like this. What about work done by non conservative. Once you release your hands and the boxes start flying this way, there's no work done by you and there's no work done by friction as well. So we're on a smooth floor. And then finally what happens is we have some kinetic energy because the boxes are moving but we have no potential energy because this spring has decompressed. So what happens is we have you elastic is going to equal the kinetic energy. Now what happens is we have two blocks that are moving. So the kinetic energy is actually gonna be one half and one V. One final squared plus one half. Em to the two final squared, the kinetic energy is made up of two objects that are moving now. So we actually have all those numbers can go ahead and calculate this. So this you elastic here. The stored energy inside the spring is one half times four. I'm sorry, this is mass one. So this is gonna be three times negative 10 squared and it becomes positive once you square it plus one half times four times um 7. squared, which end up getting is 262.5 jewels. So that's the energy that was stored inside of the spring. Let's take a look now at part C. Which is the only want to figure out how much how much the spring was compressed before launching the blocks. So we we just calculated was we just calculated the elastic potential energy 262. jewels. What we want to do now is we want to calculate the compression distance which is really just gonna be X. Initial. So we can do is we can just relate the the the elastic potential energy to one half K. X. Squared. So we're looking for here is X. Initial. And we have The elastic potential energy and we also have the spring constant. It's 800 uh newtons per meter. So we can just go ahead and I'll just go through as quickly. This is two times you elastic divided by K. And we're gonna square root that that equals X. Initial. So when you plug this in, what you're gonna get is 525 divided by 800. And you're gonna get 0.81 m. That's final answer. So these are all your answers, right? This is a use, momentum and energy conservation to solve these kinds of problems that this one guys let me know if you have any questions.