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Head-On Elastic Collision

Patrick Ford
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Hey everyone. So hopefully got a chance to work this problem out on your own. So you've got these two blocks of equal mass they're gonna undergo and head on elastic collision and we want to calculate the magnitude and direction of the final velocities after colliding. So let's go ahead and work this out one step by step. So the first thing we do is just draw a quick little sketch of what's going on. So you've got these two blocks that are heading towards each other. So I'm gonna call this one block A. And this one block B. We've got that V. A initial is equal to five. And then we've got VB initial like this is equal to negative three, right? It's to the left, so it picks up a negative sign and ultimately they're going to collide. And then afterwards we want to figure out basically where they're going and how fast. So after the collision block A and B, they're gonna have sort of unknown speeds, right? They could be both going to the right or they could be going off in different directions or to the left. So we can't draw any arrows, but basically we want to calculate what are the final velocities of these two blocks. So let's move on to the next step, we're gonna write our both of our equations for conservation of momentum and elastic collisions. Remember ultimately we want to solve a system of equations, we're gonna have to write both of these. So for the conservation of momentum, this is going to look a little bit different than this because instead of M one V one we're really gonna have M A V A. Right? That's the object. M A V A initial plus MB Vb initial equals M A V A. Final plus M A M B V B final. Right? And then let's see. So I'm just gonna and then over here we've got is we've got our elastic collision equations. Remember this is the only one that we can use. We can only use this equation for elastic collisions. And it's basically that we have V one initial plus. Actually I'm gonna call this V A initial plus V B initial equals or sorry, via final equals Vb initial plus vb final. Remember they look different +12121122 and so on and so forth. Okay, so we're basically just go ahead and start solving the system of equations by plugging in some numbers. So what happens is in the first problem here, usually we would just start plugging in the masses of each of these objects, but we actually don't know what they are but that's fine because the mass of A is actually equal to the mass of B. Because they're equal mass. Alright, so they're equal mass. So that means that Emma is equal to M B. What that really means here is we can actually cancel out the m term from the whole entire equation. Remember if you have the same number that goes through all of your terms, you can just cancel it out completely. In other words the mass actually really doesn't matter in this problem or at least in the equation. So so then let's go ahead and start plugging in some initial values. So via initial is going to be the five. VB initial is going to be the negative three. This equals V. A. Final plus VB final. And when you simplify this, what you're gonna get here is you're gonna get two equals V. A. Final plus VB final. Alright, this is the first equation that we're going to need to solve our system of equations because remember this is the one that has two unknowns. So let's look at the other equation. The elastic collision equations. Remember if we start plugging in our values via initial is five V A final is unknown. Vb initial is negative three and then VB final is unknown. Okay, so we also have these same two unknowns in this problem. And so we want to do is again we want to sort of add something to this first equation so that one of the terms will cancel out and then basically you're left with one unknown. Alright, so we want to sort of stick another equation down here um so that we can cancel out one of the terms. Okay, so what happens is when you bring the negative three over to the other side, it becomes a positive and you end up with eight and then when you move the V A final to the other side, it picks up a negative sign. So in other words you get eight equals negative V A final plus VB final. Alright, now again this is where we would have to either multiply this equation to get one of these terms to cancel out, but we actually don't have to do that here because in this equation we have a V A final and we also have a negative V A final in this equation. So we don't actually have to multiply this by anything just to make the numbers line up so we can actually just go ahead and stick this equation right down in this box. So we're gonna add this to let's see this is eight equals negative V A final plus VtB final. Notice how now when you stack these two things on top of each other and then you add them down. The V A finals will just cancel out. And what you'll end up with here is you'll end up with 10 on one side and you'll end up with two vB final on the other. Alright, so now that we've sort of eliminated one of the equations, we're just gonna go ahead and solve. So this final velocity here, this v be final for B is going to be five. So now let's move on to the last step here, which is this is one of our target variables. This Vb final here is actually going to be going off to the right like this. So this VB final equals five and we got a positive number so it points to the right. The last thing we have to do is we have to plug the first target variable into any of the other equations to then solve for the other missing variable. So in other words we can stick this VAVB final into either one of these two equations to solve for the other one. It really just your preference, they're both pretty much the same. I'm just gonna go ahead and go with the first one here. So then basically if we write rewrite equation number one, you're gonna get is the two is equal to V A final plus VB final but we actually know that that's five already. So in other words this is just gonna be five like this. Okay, so um we just go ahead and solve and we're just going to get that V A final is equal to negative three m per second and that is your other final answer. So in other words, V A final is going to be moving to the left at negative three. Alright, so now I want you to notice that something interesting happened in this problem. So we actually have these two blocks of equal mass they collided and what happens is the via initial for object A was five and then the VB initial for object B was negative three. But afterwards their velocities basically switched. So now V. A final is negative three, whereas VB final is five. So in other words, the V A. For object A became the VB the final velocity for object B and vice versa for objects for the for the second block. So in other words, when you have these two objects of equal mass and they undergo an elastic collisions. One pro tip that you can use is they actually trade or exchange velocities. So they trade or exchange, basically they just swap the initial and final velocities between the both objects. Alright guys, so that's it for this one.