Three identical train cars, coupled together, are rolling east at speed v0. A fourth car traveling east at 2v0 catches up with the three and couples to make a four-car train. A moment later, the train cars hit a fifth car that was at rest on the tracks, and it couples to make a five-car train. What is the speed of the five-car train?

Video transcript

Hey, everyone in this problem, we have a bus of mass M traveling at a velocity of V I hat, crashes into a truck of mass four M traveling at a velocity of V divided by four I hat and sticks to it. OK. So let's start there. There's some more information, but we're gonna start with that information. So let's write this out as time 0.1. OK? And in time 0.1, we have this mass of our bus which is equal to capital M. We have the velocity of the bus at this first time point of V I hat. And it's gonna crash into a truck. Now, the mass of this truck and T is equal to four M and the velocity of the truck at this initial time point T one is equal to V divided by four I have. OK. So this is everything we have at time 0.1. And we're told that these two are going to crash into each other and stick together. So after they crash, we have time 0.2 and we have the mass and we're gonna call this M BT. OK. This is gonna be the mass of the bus truck stuck together. So that total mass which is gonna be M plus four M and we aren't told anything about the speed. So that was just one sentence of this problem. It gave us all of this information. Ok. So let's keep reading and see what else we're told. Now, we're told that the bus in the truck are going to collide with a car of mass M divided by two waiting at a red light. Ok. So at time 0.2, we have this bus truck combination, but we also have a car, ok? And the mass of this car is M divided by two and it is waiting at a red light which tells us that the velocity of this car at time 20.2 is going to be equal to zero. Now, we're asked to find the velocity of the bus truck and car after the collision if they stick together. Ok. So time 0.2 is before the collision with the car. And time 0.3 will be this final time point after the second collision. So we have this mass, we're gonna call it M three. And that's when all three are stuck together, ok. The bus collided with the truck and got stuck together and then they collided with the car and got stuck together, ok. So this is gonna be equal to the total mass M plus four M plus M over two. And we're asked to find the velocity V three of these three. OK. Now they're stuck together. And so that velocity is just a single velocity, one velocity that they're all moving with. We're given four answer choices. Option A 2/5 V I hat. Option B 27 V I hat, option C 4/7 V I hat and option D 4/11 V I hat. Now we have a collision problem. We wanna think about conservation of momentum. Now, if we think about the first two time points, we know that the momentum at time 20.1 is going to equal the momentum at time 0.2. OK. The momentum before any collision is equal to the momentum after that first collision, if we look at the next two time points, we can write the same. We know that the momentum at time 20.2 is going to be equal to the momentum at time 0.3. And the momentum before the collision with the car is equal to that final momentum when all three are stuck together. No, in order to solve for this V three in our momentum time 30.3, we need to know the velocity at time 0.2. If we're just using the equation P two is equal to P three. So what we can do is we can use our first equation P one is equal to P two solve for that velocity. And then use the second equation P two is equal to P three. But we can make this simpler. If P one is equal to P two and P two is equal to P three, then we must have that the momentum P one is equal to the momentum P three. OK. So the momentum is conserved all the way through from the very first time point before any collision until this final time point where we've had two collisions. OK. So we don't even need to worry about the speed of the bus and truck at this second time point. Ok. Now, this initial momentum P one, what makes up that momentum? Now we have the bus and the truck. So we have to consider both of those. So on the left hand side, we get the initial momentum of the bus PB one plus the initial momentum of the truck P T one. And that's gonna be equal to that momentum P three. Now recall that momentum is mass multiplied by velocity. So what we get is M B multiplied by V B one plus M T multiplied by V T one. And this is equal to M three multiplied by V three. Now V three again, that's that velocity we're looking for. We want to substitute in all the other values so that we can. So now M B is going to be equal to capital M V B one is equal to capital V. And we're gonna take that positive I hat direction as our positive direction, we're gonna add the mass of the truck four M multiplied by its velocity which is V divided by four. And this is gonna be equal to M three, which is M who was for him, who was M over two multiplied by that final velocity V three that we're looking for. And now it's just a matter of simplifying and solving. OK. On the left hand side, we have M V and we have four M multiplied by V over four, a four divided by four gives us one. And so we just get plus M V there as well. On the right hand side, we have M plus four M plus M divided by two, we can simplify that into 11 divided by two M and this is all multiplied by V three. OK? Isolating for V three, V three is going to be equal to two M B. OK? Simplify the left-hand side and we're gonna divide by 11 divided by two M. Now, we can divide out the M term simplifying. This will give us four V divided by 11. And that is that final velocity we were looking for. Now, we've considered the I hat direction that's positive, but we've gotten this positive speed of four V divided by 11. And so that is going to be in the I hat direction. And so comparing to our answer choices, we see that the correct answer is going to be option D four divided by 11 V I ha thanks everyone for watching. I hope this video helped see you in the next one.