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Three-Body Collisions

Patrick Ford
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Hey everybody. So let's take a look at our problem here. So, this one's kind of interesting because we have three different objects, a bulletin two blocks and two different collisions that are happening. But basically what we wanna do is we want to calculate the speed of the second block after the bullet embeds itself into the second one. So let's go ahead and stick to the steps. We're gonna draw a diagram from before and after. So this is what's going on in this problem. We have a bullet like this, it's flying and it's gonna hit towards a block, it's gonna head towards a block. So I'm going to call this M1. This has a mass of 0.01 And the block has M2, which I'm gonna call 0.5. But what happens is the book, the bullet actually keeps on going after it's passed through the first block. So it keeps on going through the first block and actually embeds itself in a second block. I'm gonna call the M three And this equals 0.5 kg as well. And basically, once it embeds itself in there, the whole entire system is going to be moving off with some velocity. I'm gonna call this v at this point. All right. And that's really what we want to calculate the tricky thing here is that we have three different objects and two different collisions. So if you're not careful, what happens Is that if you use M1 V1, initial and MTV to initial the final will be the initial for the second collision and that can be really confusing. So, we're gonna do is we're gonna do something that we've used in other sort of chapters like motion and forces, which is we're going to label the different events that are happening. So, here's what I'm gonna do. I'm gonna call this A this one is B and this one is C. So A is actually going to be so A is the point where it's before the first collision. So this A B and C do not refer to different objects that refer to different sort of events or different points in time. I want to be really clear about that. So it's before the first collision. Now, point B is after the first collision, right? It's when the bullet exits the first block and keeps on going towards the second one. So it's after the first, but it's before the second collision. And then finally point sees when the bullet embeds itself into the second block. And so that's after the second collision. So, really what we want to calculate is this final velocity here and I'm gonna call this velocity. See that's really what we're interested in. So, how do we do that? Well, we're gonna we're gonna again stick to the steps, move on to the second thing, we're gonna write the conservation of momentum equation. If I want to calculate the VC, then I'm gonna pick the interval that basically includes, see so in other words, I'm gonna include the, I'm gonna have the interval from B to C. Right, that's the interval after the first collision, but before the second one and then when the bullet finally embeds itself. So let's set up our equation here for B- two c. So what I'm gonna do is again keep just keep in mind that we have three different objects. So really, what's going on? Is that the bullet, which is M one is going to collide with the third object, M three. So here's how our equation is going to get set up, It's going to be M one V one B, right? That's the initial from B to C Plus M three, V 3, B equals. Now, what happens is when the bullet embeds itself, that is a completely an elastic collision. So what happens here is that from B to C is completely in elastic? Whereas the first collision from A to B is just an elastic one, because nothing gets stuck together, right, the bullet exits and both of these things sort of like, you know, they go apart, right? They don't actually stick together. Okay, so this is gonna be again using our shortcut M one plus M three and this is gonna be V C. So this is really my target variable is what is the final velocity of the bullet plus block combo. So that's what I'm looking for here. So if you look through your variables, what you're gonna see is that all these masses are given to us. So can we actually just solve this by using this interval? Well, if you look at this, what happens is we're going to hit 0.01 and then V one B is going to be the velocity of the bullet after it exits the first block. Right? So this is gonna be V one B. Now, what is that velocity? If you look through the problem, you might see that this 430 m per second. But remember that's the initial speed of the bullet. In other words, the V one A is going to be 430, but what happens is after it exits the first block, it's gonna change, right? There's gonna be an exchange of velocity, there's gonna be some exchange momentum and V one B is definitely not going to be 430 m per second, but we actually don't know what that is, so I don't know what this is yet and I'm just going to leave it blank for here. Now, what about M three V three B? Well, basically this is just the initial velocity of the second block before the collision, in that case the third, the second block is stationary. So in other words, that term actually just goes away because that term is zero, So we have zero on the left side and then on the right side we have the two velocity, the two masses combined 0.01 0.5 V C. So there's only one variable that we really need and that's this V one B over here. It's the velocity of the block after it exits the first block. Or sorry, the velocity of the bullet after it exits the first block. Now, how do we go find that? Remember we in this problem, we have two different collisions, two different sort of intervals and so to go and find this V one B. I'm gonna have to go and use the other interval which is gonna be the A to B interval. Alright, so let's set up that equation now, now that equation is gonna look like M one V one A plus M. Two V two A equals M one V one B plus M2 V two B. Remember we can't use the shortcut like we did over here because this collision is not completely an elastic, the two masses will not stick together. Alright, so that's the important part. So really this V one B here is actually gonna be this V one B and that's again why we didn't use initials and finals because it can get really confusing. So let's go ahead and start plugging in the values and solving So this is going to be 0.1. What about this V one A. We actually know what that is. That's just the initial velocity of the bullet, which is the 4 30 m per second. So 4 30 plus M two V two A. So basically that's the mass and velocity of the first block of this one. And remember both blocks are at rest. So what happens here is you're just going to get 0.5 but then this cancels out because this is equal to zero. So basically the whole term goes away, Then we have a 0.01 and then V1 B. That's exactly what we're trying to find here. Plus. And then we've got 0.5. And then what about V to be? It's basically the velocity of the block after the collision. What number is that? Well, if you look through your variables, what's going to happen here? Is that the speed of the first block immediately afterwards is 5.6 m/s. So, that's basically v to be all right, that's what's going to go over here. The 5.6. So now we can do is we can solve this equation for V one B and that's exactly what we're missing in the B two C interval. Okay, So if you plugged all that stuff in, what you're gonna get is 4.3 on the left side, This is going to equal 0.01 V one b plus. And then this works out to 2.8 when you bring this over to the other side, which you're going to get here Is you're gonna get 1.5 is equal to 0.01 v. one b. And then finally v. one b. is equal to 15. I'm sorry, 150 m/s. Alright, so now that we have this 1 50, that's what we stick into this equation over here. So this becomes the 1 50 now we can go ahead and solve for V. C. This is going to be zero point uh where you work this out. This is gonna be 1.5 again equals 0.51 V. One. Sorry, VC. And now if you work this out, what you're gonna get is that VC is equal to 2.94 m per second. Alright, So it's a little bit of you know, sort of upfront work, getting organized, labeling everything. But once you do that, the rest of the math becomes pretty straightforward. All right, so that's your final answer. It's 2.94 m per second. Let me know if you have any questions