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A $5.00-\mathrm{g}$ bullet is shot through a $1.00-\mathrm{kg}$ wood block suspended on a string 2.00 $\mathrm{m}$ long. The center of mass of the block rises a distance of 0.45 $\mathrm{cm} .$ Find the speed of the bullet as it emerges from the block if its initial speed is 450 $\mathrm{m} / \mathrm{s}$ .

$v_{A 2}=$ $391 \mathrm{m} / \mathrm{s}$

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Cornell University

University of Washington

Simon Fraser University

McMaster University

{'transcript': "once again welcome to a new problem. But this time we have a bullet on the block. So given well, given a scenario where we have five gram bullets, a marshal bullet, this five grounds were going to change that right away into kilograms and multiply by one kilogram over thousand grams. And that gives us, you know, five five over thousand, which is zero point zero zero five kilograms. So this is this is a bullet that's moving that way and there's a block. There's a hanging blood right here, and the bullet is going to collide with the block. So this is kind of like the before scenario and think about this as an ox is on extended axes. You know, you could even have the y Ox is right there. The block itself has a mass off one kilogram. It's a block of wood. Okay, so we'Ll just call it a mass of wood is one point zero zero kilogram. So that's what's happening. I've also given that the initial velocity off the block is going to be four hundred and fifty meters per second. Um, so that's the information we were given. Also, given that once the bullet collides with the block, something happens. So the, um this's this is the block right here. So once once the bullet collides with the block, the block is gonna be lifted up a new height. Um, off zero point three eight centimeters. I mean, that's pretty small displacement, but we have to expand it just to show what's happening right here. So the blood gets lifted. So right now, if it happens to be here, you can see that there's a change in height for the block. So this distance right here, his zero point three eight centimetres, we can change that two meters by multiplying by, Ah, zero point three centimeters times one meter off one hundred centimetres, and this becomes a zero point zero zero three eight meters. The center, the center of gravity, rises by the same amount, zero point three eight centimeters. The bullet is going to go inside the block and it comes out on the opposite side. So the bullet emerges on the opposite side. Initially, there was on this side, and then it goes on the other side and we want to find I want to find the final velocity of the bullet. Given that the bloc will be moving, it means that it's also gonna have its Ah, remember the initial the initial velocity of the block, the initial velocity of the block, zero meters per second. And then the block also has a final velocity once you strike it. But we don't know what the final velocity is, and we also don't know what the final velocity of the bullet is, but we're going to find out. The purpose of the problem is to solve for the final velocity off the bullet as it comes out, or the speed of the bullet as it comes out. So in the next page, we know that by the law of conservation, of energy, the law of conservation, no off energy. You know, I think we have to go back. And you know what? Interested in finding the final velocity of girl off the wooden block, and that's going to help us with the law of conservation of energy. So, uh, the, uh, kinetic energy off the wooden block. Um So the initial Connecticut do, plus the initial potential energy equals to the final kinetic energy off the old block, plus the final, Uh, a potential energy off the old block. We know that the initial height zero So this one is just gonna be zero thie Final kinetic energy also is going to be zero. Because as the block rises, its final velocity will be zero. So these to a zero. So then we have one half mass off wooden block velocity. Her initial velocity off wouldn't lock square because two massive wooden block G and final height of you didn't look. And so we're supposed to solve for this. This velocity was supposed to soften for this velocity. Um, I think we have to change some numbers here. The final velocity of the wooden block is zero meters per second. What we don't know is the initial velocity of building blocks. So that's what we need in this case. Remember it? Swing. So initially we don't know what that velocity is, but we know that it's going to swing up until it gets to a certain height, and then it stops. So we need to find this initial velocity and so way saw for that, the ice squared equals to multiply both sides by two and then divide both sides by the Marcel's, the wooden block thes to cancel out on these to cancel out those to cancel out. And we're left with G final height twice and saw the initial velocity of the block is just worry called G the final height. And remember the final height is this one right here? This is the final height of the block, so we have that and then in the next page will use the law of conservation off mo mentum. Our goal is to find the final speed of the bullet. The law of conservation Off momentum say's that the initial momentum equals to the final momentum. The initial momentum involves the bullet and the block. So the mass of the bullet times the initial velocity of the bullet plus the mass of the wooden block time's thie times the initial velocity of golden block. And this is, you know, before it it it it Ah, um it starts moving, so they're two components. I think we should call this before motion initial velocity before motion. And this is an initial velocity after motion. And so we have initial velocity one and initial velocity to that's for the wooden block you know, want to be very careful about that process. And so this one was actually the initial velocity to, you know, this is the initial velocity after collision. So after they collide, you know, that's going to be the initial velocity. So, you know, we go back and say the initial the fast initial velocity. This is obviously going to be zero. As you can see, if you go back, you see that this is zero and then make that equal to the mass of the bullet. You know, we're looking for this motion there. So massive bullet and the final velocity of bullet after it comes out. And then we have the mass of the wooden block a times the initial velocity off the road and block. But this is the second initial lost, not the fast initial plastic. So you have to make that distinction. Our goal is to solve for this number right here. So we're going to move this one to the left. We're going to move the more momentum off the wooden block when it's initially moving to the left. And so we have muss m bvd final equals two massive bullet velocity of bullet initial minus myself wouldn't look The lost years wouldn't lock initial This are the two Momentum's was subtracting them from each other on then. Of course, the next partof The problem is that, um we'll divide we'LL divide everything by we'Ll divide everything by by the mass of the bullet So by the mass of the bullet like that Okay, so we're dividing everything by the mass of bullet and then on the next page to get the final So just want to make sure final velocity of the bullet becomes equal to notice. These two numbers are going to cancel out the mass of the bullet in this massive bullets All we're left with a simplified part of the problem. Vv i minus Marcel's wouldn't lock of a Marcel's bullet times the second initial velocity of the block when it's swinging member initially, the block is like right there. Okay, so it's initial velocity before it swings is just zero meters per second and then when the block starts to swing, it has kind of like another initial velocity because eventually, once it swings is going to come to a stop. Okay, so this is kind of like it's final, our velocity and this is the initial when it starts to swing, that's what we're calling it. Subscript two, this is the initial ah, before any impact, you know, before any influx of the college One next thing, we plug in the numbers. So we have four hundred and, um fifty. So this one is this one is going to be the initial lost of the bullet minus. I am doubly off a m B. We did so for the second initial velocity off the bullet off the block, which is to G. Paige, if radical. Okay, final. And so you know, we have four hundred and fifty meters per second, minus one kilogram over the mass of the bullet, remember was a zero point zero zero five kilograms and then radical two times nine point eight meters per second, squared times the height, which happens to be zero point zero zero three eight meters. So that's that's what we have right there. And then one was simplified. The problem is, involves a lot of decimal, so you have to be very careful about all the steps Wei have. Let's see how one divide by point zero zero one day quite by point zero zero five and then multiply that by this particle, which is too times nine point eight times Princess Rosa, create bones. Then we have to subtract four hundred and fifty. So we get the final speed off. The Buller becomes equivalent to three hundred and ninety five point for two meters per second. So this is this is what's happening to this problem. Just as every cup, you know, we have a block and a bullet. A bullet is fired through the block. It pushes it up. So the bloc gained some type of mental kinetic energy becomes potential and a few final potential energy. We use that to simplify the initial velocity off the block. When it starts to swing up off course, the final velocity of the block where it's swinging is going to be zero on. Then we use the law of conservation, of momentum, momentum of the system for collision and then after collision. And that gives us the final lost hope. You enjoy the problem. Feel free to ask any question, send questions my way and have a wonderful day. Okay, Thanks. Bye."}

California State Polytechnic University, Pomona