When cars are equipped with flexible bumpers, they will bounce off each other during low-speed collisions, thus causing less damage. In one such accident, a 1750-kg car traveling to the right at 1.50 m/s collides with a 1450-kg car going to the left at 1.10 m/s. Measurements show that the heavier car's speed just after the collision was 0.250 m/s in its original direction. Ignore any road friction during the collision. (a) What was the speed of the lighter car just after the collision? (b) Calculate the change in the combined kinetic energy of the two-car system during this collision.

Video transcript

everyone in this video, we have two cars driving towards each other who are going to collide and we want to find out what is the change in total kinetic energy of the two cars, after the collision. Okay. And we're told to ignore friction. Alright, so the first thing we want to do with a collision problem like this, is to go ahead and draw the system both before and after the collision. Before the collision, we have two cars, This red car, you can pretend this box is a car and this blue car. Okay, now we're told the red car, or the first car has a mass of 900 kg and that it's headed east at 1.4 m per second. Okay, So we're gonna say east is to the right, and what we're gonna do is we're gonna label the right to be the east and to be our positive X direction. So, we were talking about the velocity of this car again, it's speed is 1.4 m per second East. We're taking east to be positive, so the velocity will be positive here of car 11. m per second. Now, part two. Car two is our blue car. We're told at 650 kg not that into our drying room, and it's headed west at 1.2 m per second. Okay, so West is going to be to the left, East is to the right, West is to the left And now, because it's traveling left or west towards the opposite direction of what we've called our positive direction, the velocity of Car two is going to be negative 1. m/s. Alright, Now, after our collision again, we're gonna have our two cars. We'll draw our blue car here. Whoops, 600 50 kg and a red car. 900 kg. And we're told that this first car, the 900 kg car is moving .18 m/s in its original direction. So it's still going to be going to the right. The velocity will still be positive. Okay. And it will be going 0.18 m/s. And then our blue car is 650 kg car while we're not told anything about its velocity. Alright, so what are we asked to do? We're asked to find the change in total kinetic energy. Okay, well, the change kinetic energy is just equal to the final kinetic energy minus the initial kinetic energy. So, this is what we're looking for here, delta K. Change in total energy kinetic energy. Alright, let's work out some of these details. We're gonna kill that cannot. It's OK. Not. Ok. Is our initial kinetic energy. Okay, well, we have two cars are kinetic energy is going to be made up of the kinetic energy of car one plus the kinetic energy of car two. Okay. So. Okay. One, not plus que tuna. Now, recall kinetic energy is given by 1/2 MV two. So, for each of these we're gonna have one half corresponding mass corresponding speed squirt. 1/2 corresponding glass corresponding speed squirt. Okay, substituting the values we know We get 1/ Times 900 kg Times 1.4 m/s squared. OK, we're just going to add on the second row. So we have room to write our units kg. Okay, and the velocity of car two initially is -1. m/s. Now, when we're talking about energy, the direction doesn't matter. Um So we don't need to include the negative but we're squaring it. Okay, so either way, whether you include the negative or not, you're going to get the correct result. Okay, now this is going to give us if we do the math, it's gonna give us 1350 jewels. Okay, now, where did the jewels come from? Well, we'll do a little aside here. So here we have kilogram that we're multiplying by meters per second. All squared. So that's gonna give us kilogram meters squared per second squared. Okay, well, we can write that as kilogram meter per second squared times meter. Well, kilogram meter per second squared is newton. So we get newton meter, which we know is equal to a jewel. Okay, so we end up with the jewel, which is the unit for energy. So that's the proper unit. So that's good. Alright, now let's do the same thing but with K. Final. Okay, so, Okay, final. Again, we have two cars. So we have to consider the kinetic energy from the first car and the kinetic energy from the second, car one half M. The squared for each term 1/2. M two V 2 F Squared. Well, we run into a problem here, we don't know, be too bad this value from the blue car. So in order to determine our change in total kinetic energy, delta K. We're gonna need to find V two F First. Okay, well, let's consider the momentum. They were told to ignore friction so there's no net external forces. So we know we have conservation of momentum. Conservation of momentum tells us that the momentum in the system initially is going to equal the momentum in the system at the end. After the collision again, our system has two cars. So the momentum is going to come from Car one. So we have the momentum from car one plus the momentum of Car two. That's both. Before and after the collision recall, momentum is mass times velocity. So for each of these terms we have mass times corresponding velocity M to me. Do not. M one. V 1, F M two. V 2. Now substituting the values we know we know M1 is 900 kg V one not is 1.4 m/s. M - 650 kg. The two not is minus 1.2 m per second. And here momentum has a direction. Okay, so with momentum we have to consider the direction. Just like velocity. So the momentum equation has velocity term in it. So we need to include the negative here to indicate the direction and that is going and sorry, in this equation above this, plus, in the middle here should be an equal, not a plus. Okay, equals Alright, so then we have equals 600. Not so common. 900 kg. Okay, the velocity of car one after the collision, 0.18 m/s. 650 kg for Car two and V two F. Its final velocity. That's what we're trying to find. Okay, Alright, so working out all of the math here, on the left hand side, we're gonna have 480. And the unit there will be kilogram meter per second. Okay, And on the right hand side we're gonna have 162, kg times V two. And the units on the 162 it's gonna be the same. It's gonna be kilogram meters per second. Okay, So we can go ahead and move that over. We get 318 kg meters per second, Is equal to 650 kg times v two f. And then dividing the unit of kilogram will cancel. And we'll be left with V two F Is equal to 0.4892, m/s. That's the proper unit for velocity, so that's great. That works out. Okay, so now we have our V two F. What are we trying to find? We're trying to find that delta K. Total change in kinetic energy in order to do that. We need k F. Okay, so now we can go ahead and plug in the V two F we found. So here we're gonna have one half M 900 V one F. Squared. Okay, so the velocity of car one at the end 0.18, that's one half 650 times is V two F. We just found 0.489 squared. Okay, And the units here haven't written them, but they're gonna be the same as the units above. So we're gonna have kilograms times meters squared per second squared. And again when we work all of that out, we end up with the unit of jewels. Okay, so if we do all of this, we're going to get 14 kg meters squared per second squared plus 77. kg meters squared per second squared. Okay, so that's gonna be 92. jewels. We're just keeping some extra digits there, so we don't have any round off there. Okay, so we're in jewels. Alright, so we found we found k final, we can finally find that delta K. We're looking for. Okay, so it's just gonna be K. F 92. 45 -1350. Okay, those are both jewels, jewels. And so our final answer is going to be one minus And 57.63 jewels. Okay, and just rounding to the nearest whole number, we're going to get minus 1,258 jewels and that's going to correspond with answer D. Alright. Thanks everyone for watching. I hope that helps you in the next video.