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California State Polytechnic University, Pomona

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Problem 32 Easy Difficulty

A $1050-\mathrm{kg}$ sports car is moving westbound at 15.0 $\mathrm{m} / \mathrm{s}$ on a level road when it collides with a 6320 $\mathrm{kg}$ truck driving east on the same road at 10.0 $\mathrm{m} / \mathrm{s}$ . The two vehicles remain locked together after the collision. (a) What is the velocity (magnitude and direction) of the two vehicles just after the collision? (b) At what speed should the truck have been moving so that it and car are both stopped in the collision? (c) Find the change in kinetic energy of the system of two vehicles for the situations of part (a) and part (b). For which situation is the change in kinetic energy greater in magnitude?

Answer

(a) $v_{2}=6.44 \mathrm{m} / \mathrm{s}$ to the east.
(b) $v_{B 1}=2.49 \mathrm{m} / \mathrm{s}$ to the east
(c) $\Delta K_{(a)}=-281377 \mathrm{J} \quad-\quad \Delta K_{(b)}=-137750.2 \mathrm{J}$

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Video Transcript

{'transcript': "Yeah. Once again, welcome to a new problem. This time we have a truck and a sports car. So the sports car were given that the sports car is moving in the, uh towards the left eastward, so that means it has a negative velocity. The mass of the sports car happens to be given us 1000 and 50 kg, and it's moving with an initial velocity, um, towards the left, which is negative 15 m per second. That's the sports car. And then we also have a truck. Well, given that you know there's a truck and it's moving towards the right, the mass of the truck happens to be, um, 6000. So the mass of the truck happens to be 6000 320 kg. The velocity initial velocity of the truck is given us mhm 10 meters per second. So that's what's going to happen. That's the information we're given its positive because it's moving towards the right. Um, we want to find in part a the a final velocity when the truck and the sport's got collide with each other. So you know they're gonna collide and and stick together. Um, and we don't know what direction they'll go, but, you know, you can assume they're going towards one direction. Uh, and we want to find that final velocity when the collision happens. Uh, the second thing we want to find out. So this final velocity we want to find the magnitude and direction. Those two things you want to find out is going to go to the left or will it go to the right? The second thing we want to find out in part B is the speed that the truck is moving. Um, so the initial speed of the truck such that Okay, such that, uh, they stop on collision. So another way of framing the problem would be if they both have an initial, uh, the same the same momentum. Um, when they're moving towards each other, they're gonna stop, and that's what be. And then in Patsy, we want to find what we call the or the change in kinetic energy for the two parts. So, you know, what's the what's the change in kinetic energy in part? A. And then what's the change in kinetic energy? Uh, in part B. So that's what we want to find out. And then the final thing, Uh, the asking is which one is bigger? So is the change in kinetic energy in part A Mm. You know, Is the change input a greater than in part B, for example? You know, that's that's one thing you want to, uh you want to check? Okay. Yeah. Mhm. Yeah. Mhm. Yeah, sort of. Yeah. So, um, that's the information we're given in the problem. And, uh, you know the first part The first part of the problem is that we are going to compute the final velocity using the law of conservation of momentum, using the law of conservation of momentum. We get to see that the initial momentum, it goes to the final momentum. We can switch that because we are targeting the final velocity. So they're sticking together. So the moss off the truck, plus the mass of the car and the A final velocity is the same as, um is the same as the mass of the truck and the initial velocity of the truck. Uh huh. Plus the mask of the car, times the initial velocity of the car. And so, you know, moving things around to see that the final velocity is the initial momentum. Mhm of the car, plus the final momentum off the car over the two. Okay, the sum of the two masses together. And so you know, we'll we'll do this in the next page. So the final velocity becomes 6320 kg times. That's the momentum of the truck. 10 m per second, plus the momentum of the car. The sports car, um, times negative. 15 m per second. You want to divide that by the some of the masters, which is six, 6320 kg plus 10 50 kg. Um, and so you know your final answer become six point 43 me this second, and it's positive. So the if you go back, we're looking for the magnitude and direction. So it means that the, um two, uh two. Yeah, vehicles will move towards the right wind. A common velocity. Oh, 6.43 So we'll see 6.43 m per second. You know, that's what's going to happen to that. Uh, the second part you can see, we were We wanted to find the speed such that the, um So they want to find the speed of the truck if they happen to collide and stop. So at what speech? Had the truck been moving so that they both, uh, they stopped after collision. So we'll do that on the next page. So in terms of what we're saying is we have a truck and then we have a car. Um and, you know, they were moving towards each other, they collide, and then they stopped. So if they collide and stop, that just means that the momentum, which is the mass of the truck and the initial velocity of the truck, has to be equal to the mass of the car. Um, and the initial velocity of the car. So we don't know what this velocity is. Uh, but we we can solve it. We put those two together. Remember, this is the outcome. When they stick to each other, it means the momentum's are the same. Um and so, uh, this turns out to be the mass of the car is 10 50 kg and its initial of lost eight towards the left is negative 15 meters for second. And and the mass of the truck is huge. It's 6000 320 kg. That just means that the new velocity happens to be negative 2.49 m per second. Uh, it's you gonna be moving towards the left. So if you go back, we were saying, you know, what's the speed such? That they're going to stop? You know, that's what you're looking for. Speed has no magnitude or speed has a magnitude no direction. So it's just gonna be 2.49 m. Second, that's the speed of the truck if they have to stop if they have to stop when they collide. So it means that you know, the the, uh, in terms of velocity the truck will be moving, You know, that week before they stop. Um, and that just comes from the arrangement organization of the problem. The last part of the problem, which is Patsy. We're going to do that in the next page and was saying that you know, what's the change in kinetic energy in the two parts? What's the change in kinetic energy? Remember, this is what happened in part a. So change in kinetic energy is, uh, the final kinetic energy of the system, minus the initial kinetic energy of the system. The final kinetic energy of the system. Remember that when they collide, they stick to each other. So that's M C. And the final squared minus the initial, which is before they stick to each other. This is the initial momentum of the truck. Um, and then plus, you know, we need a parenthesis there just to make our life easier, plus the initial momentum of the car like that. And so, uh, when we plug in the numbers, we get one half, 6000 320 kg, plus the mass of the car is 10 50 kg times the final velocity that we computed was 6.43 m per second. Um, and if you can recall, it was actually positive. So 6.43 m per second square. That and then we subtract. One half mass of truck is 6000 320 kg, and the velocity of the truck initial velocity of the truck, you can see was, um uh, the initial velocity was 10 m per second, and you still have to square that because we're dealing with, um, kinetic energy in this case. So 10 m per second squared, and then also we have plus one half mass of the truck is 10. 50 kilogram. The last year of the car is if we go back. He was negative. 15. So we have to include that. Meet us second squared. Um, and so the change in part A would be a positive to 81.7 times 10 to the three. Jules, you want to call that? Mm. You know, that's the change in kinetic energy. They're asking us to compute that find the change in kinetic energy of the system of the two in in both the situation. So, uh, this is the change in kinetic energy For the first part, which is a two 81 0.7 times 10 to the three. Jules, the second part, we have to get the change in kinetic energy once again. You know, it's the final, uh, minus the initial kinetic energy. They stick together. So the final kinetic energy zero jewels. Um, and we want to find the initial kinetic Kennedy before they stick together. So one half, uh, m t v t initial squared, plus one half m C VC initial squared. Those are the initial and final momentum in that case. So we have one half times the mass of the truck. 6320 kg. Always a good habit to have the units there. Initial of the law. Steel of the truck. Remember, this is the velocity that's going to make them stop when they when they collide with each other. And then this is plus one half the mass of the truck is 10 50 kg. The initial velocity in that case was negative. 15 m per second squared. When you simplify that, you get negative 1 37.7 times 10 to the three jewels. You know, that's the change in kinetic energy in the second case. Okay, Mhm. And so we can conclude that, um, you know, as they were asking as part of the problem, you know, if you go back, you see that the change in kinetic energy in the first case, um, is greater than in the second case. So, in terms of magnitude, so we'll say the magnitude of kinetic energy in part a mhm, uh, is greater than in part B. Okay, Hope you enjoy the problem? A couple of steps going on the first part. We're finding the final velocity when they stick together. 6.43 m per second. The second part. We're finding the speed of the truck. The ideal speed of the truck for them to stop 2.49 m per second for the truck. This is the last day of the initial velocity of the trunk, as we call it. Well, actually, this is the initial blast in the truck. And then, you know, you can change that to speed. Speed is a scalar quantity. Um, and then finally you compare The energy is in part a n b. And you can see that the energy part is bigger. Hope you enjoy the problem. Feel free to ask any questions, uh, and have a wonderful day. Okay, Thanks. Bye."}

California State Polytechnic University, Pomona
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