Momentum collision graphs

A cart of mass m moving rightward at speed 2v hits a slower moving cart of mass m moving rightward at speed v. When the carts collide, they hook together. There's friction between the track and carts and between the moving parts of the carts. Which of the following best represents the momentum of the left side cart as a function of time before and after the collision?

So before we even look at the choices, let's see if we can sketch it out ourselves. So pause this video and see if you can make a sketch of what the momentum of the left side cart would look like as a function of time. And actually try to do it for the right side cart while you're at it.

Alright, now let's do this together. So just to make sure we understand what's going on, this cart is moving too. They have the same mass. This left cart is moving to the right with the velocity of 2v. The cart on the right has the same mass and it's moving to the right with half the velocity. Since momentum is mass times velocity, this will also have half the momentum.

So one way to think about it is if on our vertical axis—let me draw a straighter line—if on our vertical axis we have momentum, and on our horizontal axis we have time. So this is time right over there.

So this left cart will start with some momentum, but since there's friction in the system, especially in the mechanics of this cart, you're going to see it slow down. The force of friction is going against the direction of motion, so it's going to decelerate it. Then it's going to hit the other slower moving cart.

And that other slow... one way to think about it is some of that momentum will get transferred to the slower moving cart because then they're going to be moving together at the same velocity. You could imagine—and then at that slower velocity, it will continue to decelerate because of the friction.

So the graph of momentum over time of the left cart might look something like that. Now just for kicks, if you want to think about what about the right cart? Well, the right cart has the same mass but half the velocity, so it's going to have half the initial momentum. And so it would also degrade. It would look something like this.

But then right at the collision, the right cart would gain momentum and then they should go together like that. So these are the lines for each individual one. If you wanted a combined system, well then you would add these momentums, and you would have something that just would be a line that degrades.

But anyway, they asked us about the left cart and so we said, "Hey, something that looks like this should work out." So let's look at the choices.

So this first choice has two problems with it. First, it shows a world in which you have no friction; it's just the momentum stays constant. And so that means its velocity to the right is staying constant. Then when it's trying to show the collision, it makes the momentum negative, and this would imply that all of a sudden that cart would be moving to the left where the convention that we typically use is right is positive and left is negative. So we can rule this one out.

This second one looks a little bit better, but once again, it's assuming a world without friction where the velocity stays constant, and so the momentum stays constant until you have the collision. Then you still have it after the collision; the momentum stays constant. So we can rule that out. If they said it was frictionless, this is what we would have picked.

Choice d is looking good in the beginning, but then it shows that the momentum goes to zero, implying that the left cart just comes to a standstill. Once again, we would not expect that, so we would rule that out.

And so this is very close to what we drew—choice b.