11. Momentum & Impulse
Completely Inelastic Collisions
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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 3 different objects, a bullet and 2 blocks, and 2 different collisions that are happening. Basically, what we wanna do is we wanna 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 I'm gonna call this m one. This has a mass of 0.01 and the block has m 2, which I'm gonna call 0.5. 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 3, and this equals 0.5 kilograms as well. And, basically, once it embeds itself in there, the whole entire system is gonna be moving off with some velocity. I'm gonna call this v at this point. Alright? And that's really what we wanna calculate. The tricky thing here is that we have 3 different objects and 2 different collisions. So if you're not careful, what happens is that if you use m 1v1initialandm2v2initial, the final will be the initial for the second collision, and that can be really confusing. So what 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 gonna 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. They refer to different sort of events or different points in time. I wanna 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 c is 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 c. That's really what we're interested in. So how do we do that? Well, we're gonna stick we're gonna, again, stick to the steps, move on to the second thing, and we're gonna write the conservation of momentum equation. If I wanna calculate the v c, then I'm gonna pick the interval that basically includes c. 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 to c. So what I'm gonna do is, again, keep, just keep in mind that we have 3 different objects. So, really, what's going on is that the bullet, which is m one, is gonna collide with the 3rd object, m 3. So here's how our equation's gonna get set up. It's gonna be m one v one b. Right? That's the initial from b to c. Plus m 3v3b equals now what happens is when the bullet embeds itself, that is a completely inelastic collision. So what happens here is that from b to c is completely inelastic. Whereas, the first collision from a to b is just an inelastic one because nothing gets stuck together. Right? The bullet exits and both of these things sort of, like, you know, they they go apart. Right? They don't actually stick together. Okay. So this is gonna be, again, using our shortcut, m 1 plus m 3, 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, which 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 gonna hit 0.01 and then v one b is gonna 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 meters per second, but remember, that's the initial speed of the bullet. In other words, the v one a is gonna be 4:30, but what happens is after it exits the first block, it's gonna change. Right? There's gonna be an exchange of velocity there. There's gonna be some exchange of momentum, and v one b is definitely not going to be 430 meters per second. But we actually don't know what that is. So I don't know what this is yet and I'm just gonna leave it blank for here. Now, what about m 3v3b? 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 0. So we have 0 on the left side and then on the right side we have the 2 velocity the 2 masses combined, so 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 1st block or, sorry, the velocity of the bullet after it exits the 1st block. Now how do we go find that? Remember, we in this problem, we have 2 different collisions, 2 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 m 2v2b. Now remember, we can't use the shortcut like we did over here because this collision is not completely inelastic. 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 in finals because it can get really confusing. So let's go ahead and start plugging in the values and solving. So this is gonna be 0.01. 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 meters per second. So 4:30 plus m 2v2a. So, basically, that's the mass and velocity of the first block of this one. Now, remember, both blocks are at rest. So what happens here is you're just gonna get 0.5, but then this cancels out because this is equal to 0. So, basically, the whole term goes away. Then we have a 0.01 and then v one b, that's exactly what we're trying to find here, plus. And then we've got 0.5. And then what about v 2b? It's basically the velocity of the block after the collision. Now what number is that? Well, if you look through your, variables, what's gonna happen here is that the speed of the first block immediately afterwards is 5.6 meters per second. So that's basically v 2b. Alright? That's what's gonna go over here, the 5.6. So now what we can do is we can solve this equation for v one b, and that's exactly what we were missing in the b to c interval. Okay? So if you plug all the stuff in, what you're gonna get is 4.3 on the left side. This is gonna equal 0.01vonebplus, and then this works out to 2.8. So when you bring this over to the other side, what you're gonna get here is you're gonna get 1.5 is equal to 0.01voneb. And then finally,voneb is equal to 15 I'm sorry, 150 meters per second. Alright? So now that we have this 150, that's what we stick into this equation over here. So this becomes the 150, and now we can go ahead and solve for v c. This is gonna be 0 point, we work this out. This is gonna be 1.5 again equals 0.51v1 or, sorry, v c. And now if you work this out, what you're gonna get is the v c is equal to 2.94 meters 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. Alright? So that's your final answer. It's 2.94 meters per second. Let me know if you have any questions.
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