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Intro To Elastic Collisions

Patrick Ford
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Hey guys. So in some problems you'll see that objects collide but they undergo what's called an elastic collision. Remember that? This is one of the two sort of broad types of collisions that you could have. What I'm not showing this video is that like all collisions, we're going to solve these kinds of problems by using conservation of momentum. But for elastic collisions we're also gonna need a special extra equation to solve them. Let's go ahead and check this out. Remember that? The whole idea is that momentum is always conserved regardless of the type of collision, whether it's elastic or inelastic. But the elastic collisions have a special sort of characteristic, which is that they also conserve kinetic energy. So the kinetic energy is also conserved. So K initial is K final for the system. So we're gonna use this characteristic right here to actually help us solve our problems. Let's go ahead and work out this example together, we'll come back to this in just a second. So the idea behind this problem is that you have these two blocks that are smashing into each other. You can kind of think of this as like two billiard balls that are crashing into each other. So we have this one that's going to the right, this one's going to the left, and we want to do is we want to calculate the final velocities of both the blocks after the collision. So we're gonna go ahead and stick to our steps. We're gonna need diagrams for before and after. So let's go ahead and draw that right. So afterwards, after these things collide, we're gonna have this three kg box, this five kg box, and we want to figure out their final velocities. So if I call this object one and two, what we're really looking for is V one final and then V two final. So basically these are our target variables like this and that brings us to the second step. We're gonna have to write our conservation of momentum equation and we'll come back to this in just a second here. So our conservation of momentum is M one V one initial plus M two, V two initial equals M one V one final plus M two, V two final. Right? So we have our masses and some of the speeds, let's go ahead and start plugging in our numbers. So we have five plus three equals five plus three. And now we have the speeds right? So this five kg block is initially going to the right at two. So I'm gonna plug into this one is going to the left at four, so I'm gonna plug in negative four. And then these two final velocities are actually our target variables. Remember this is what we're looking for here. So I can simplify the left side because I have all the numbers, This is just 10 -12. So you get negative too, so negative two equals five. V one final plus three V two final. So we still have two unknowns on the right side of this problem here. Right. And you can't assume that the V one final, V two final are the same because that doesn't necessarily happen in elastic collisions. So what happens is whenever we end up with an equation with two unknowns, we're gonna need another equation to solve it. And that's what's special about elastic collisions. For elastic collisions only, we often must use an extra equation, which I like to call the elastic collision equation. It goes like this V one initial plus V one final equals V two initial plus V two final. This really is sort of like the mathematical consequence of this conceptual point right here, which is that the kinetic energy is the same. Your textbooks will derive this, but you don't really need to know the details. So all you need to do is just memorize this equation. And there's a couple of really important things about this equation, which I like to talk about here. So the first has to do with sort of like the order of the variables and how it looks similar to the conservation of momentum equation. Here's what I mean. So you're conservation of momentum has the ems and it also has 1212 It goes initial, initial, final and final. The elastic collision equation has no masses. And if you look at the letters and the numbers, it goes 11 and then initial, final, initial final. So the order of the variables is different. What I want you to remember here is that conservation of momentum goes and the elastic collision equation goes 11 2 2. That's a really easy way to remember it. Right? So the second thing I want to point out here about this equation is that this equation actually has the same unknowns as the one that equation that we got stuck with. So what happens is this equation here also has V one final and V two final here. So because we have two equations that have the same unknowns, we're going to end up with what's called a system of equations. Remember we have system of equations, there's a couple of different ways we can solve for them. And the easiest way to solve this is by using equation addition. So, here's what I'm gonna do. I'm gonna write up my mic elastic my elastic collision equation over here. So I'm gonna have V one initial plus V two initial. Sorry, it's the one final, so V one final equals V. Two initial plus V. Two final. So right on my elastic collision equation, What I want to do now is I want to solve this system of equations by using equation edition. Remember that equation edition, you're gonna line up the equations top to bottom so that you can cancel out one of the variables and eliminate one of them. If you eliminate one of the unknowns, you're gonna only gonna be left with one unknown variables. What I want to do is that basically want to put another equation down here so that I can add them together and eliminate one of the variables. So what I wanna do is I want to start off with this elastic collision equation and I eventually want to get it down here in a form where I can add them and then cancel out one of the variables. So let's go ahead and take a look here. We know these initial velocities. So I know this view in initial is to so two plus V one final equals and this is just the negative four, right? This V two initial. So this is a Plus V two final. So if you take a look here, I have a number and then V one final and then V two final. So I want to make this equation look the same way. So I'm going to move the numbers to the left and I'm going to move the V one final to the rights which end up with is six equals negative V one final plus V two final. So notice how I'm almost done because I have a number of the one final and a V two final. The problem is if I just plug it in the way it's written, I'm not gonna be able to cancel one of the variables because I have five and a three attached to these variables and I have no numbers here. So I'm gonna have to do is I'm going to have to multiply this equation by some numbers that I can get rid of one of the variables. Hopefully you guys realize that I eventually want to cancel this negative V one final and this positive VV one final because they're opposite signs. When you add them together, one of them is gonna cancel. So what I want to do is I want to make the number over here, be the same as the number over here. So we're gonna have to multiply this number by five. And when you end up doing is when you multiply this whole entire equation by five on the left side, you're gonna get 36 times five equals this becomes negative five V one final, right? Plus five V two final. So hopefully you guys realize that now when we add these two equations straight down from top to bottom, The V one finals are going to cancel out. And so all you're gonna be left with is the V two final, which you end up with is 28 on the left side and then eight V two final. And so when you go ahead and saw for V two final, you're gonna get 3.5 m per second and that's one of your target variables. So that is the final velocity of this three kg block. It's going to the right At 3.5 m/s. So now we figured out one of our target variables, how do we figure out the other one? This V one final. Well, remember that whenever you saw the system of equations and find one variable, you're gonna plug that first target variable into your other equations to solve for the other targets. You're basically going to plug this number into either one of these equations. It doesn't matter which one you use to get the one final. So I'm gonna start off with this one over here because it's the easier one, the numbers a little bit nicer. So I'm gonna bring this down here and I'm gonna have a negative two equals five V one final plus three. And I actually know what this V two final is. I just saw for that, it's 3.5. So now what happens is when you move all the numbers around, you're gonna get negative 12.5 equals five V one final. And what you'll get when you solve is V one final is equal to negative 2. m per second. So, notice how you got the other target variable by plugging into either one of your equations. So now this five kg block is actually moving to the left final and its velocity is negative 25 meters per second. All right. So that's how you solve. You just write your conservation of momentum, your elastic collision equations. You're gonna solve a system of equations for those two unknowns. All right. So the last thing I wanna point out here is that you might actually not see this elastic collision equation written like this in your textbooks. Sometimes the orders will be different. You might see some minus signs or something like that, but I highly recommend that you or that you memorize it this way, because I think it's the easiest way to learn. All right, that's it for this one, guys, let's move on.