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## AP®︎/College Physics 1

### Course: AP®︎/College Physics 1 > Unit 4

Lesson 4: Representation of changes in momentum# Changes in Momentum Worked Examples

Learn about how the momentum between two objects changes when they interact. These worked examples show how momentum is conserved when two objects collide with one another and how an external force can change the total momentum of a system.. Created by David SantoPietro.

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## Video transcript

- [Instructor] So, here's a pink ball rolling toward a green cube that's sitting at rest on
a frictionless surface. When the pink ball hits and
slams into the green cube, it's gonna exert a force to
the right on the green cube and the green cube's gonna speed up. But because of Newton's third law, whatever force the pink ball exerts on the green cube to the right has to be equal and opposite to the force the green cube exerts
backward on the pink ball. So, the green cube's
gonna gain some momentum, but the pink ball is
gonna lose some momentum. And here's the cool thing. Whatever momentum is
gained by this green cube has to be the same amount of momentum that's lost by the pink ball. So, if the green cube gained three units of momentum to the right, and then, the pink ball has to lose three units of momentum to the right. And this is true, whether these are the same
mass or different masses, or whether the green cube was moving left to
start or right to start. However they collide, if two objects collide, the change in momentum
between the two objects has to be equal and opposite. And this is why we love momentum. Whatever momentum is gained by one has to be lost by the other. And so, how is that possible? You might be like, "How can that work?" Well, let's go up here. Think about these forces. The forces on each object
are equal and opposite. Here's the cool part though. The time they're in contact
also has to be equal because as soon as one of them
loses contact with the other, the other loses contact with the one. And in physics, force times
time is called the impulse. Now, this is the change in momentum. This will equal the change in momentum. So, if you wanna know how much momentum was gained by this green cube, multiply the force
exerted on it by the time, you'll get the change in
momentum of the green cube. That's true for this pink ball. And look, the force is backwards. It's just the negative
of this other impulse. So, this is why the pink ball, no matter what it's mass
or how fast it's going, same with green cube, these changes in momentum
must be equal and opposite simply due to Newton's third law. So, to be clear, what I'm saying is this. If you are to graph the momentum, this pink ball would have
started off with some momentum. And that would have been
constant before the collision if there's no friction or resistance. But then, it lost some momentum. So, this goes down and
maybe end up right here. And then, afterward, it'll maintain a constant amount of momentum. The green box, the green cube here, started with zero
momentum, it was at rest. During this collision,
it gained some momentum. So, it's gonna jump up maybe to here. What I'm saying is that, and
afterward, it stays constant, what I'm saying is that
these two jumps are the same. You know, if this thing lost, let's say it lost four units of momentum, well then, this green cube has to gain four units of momentum. And you might be like,
"Okay, I don't care." But here's why you should care. If you are to graph the total momentum, what this means is that
the total momentum, which was just the pink ball initially, is gonna remain constant the whole time. It's as if the momentum never
realized a collision occurred. This total momentum just remains constant. And this is why conservation
and momentum is a thing. It is why it's really useful. The total initial momentum in a system, even if a collision occurs
between objects in that system, the total initial momentum must be equal to the total final momentum. As long as the forces are only internal, that is to say between
objects in the system, and there's no external forces, then, this will always be true. So, this is a super powerful tool we can use to problem-solve. This saves us a lot of time and trouble. If a collision is occurring, this is one of our best
methods to solve for things is conservation of momentum. Now, keep in mind. Even though the total momentum in a system has to stay the same, the momentum of an individual object does not have to stay the same. These objects can exchange
momentum, but again, the reason this is gonna be conserved is that they do so equally. If one gains five, the
other loses five and so on, and the total amount stays the same. So, let me show you how
this works real quick. So, to give you an idea numerically here. So, let's say the pink
ball was two kilograms and it was going five
meters per second start. And the green cube had a
mass, m, and afterward, let's say the pink ball's
going four meters per seconds. So, it's slowed down. And the green cubes speeds up. Let's say, it's going
eight meters per second. You might think, "Wait,
this isn't conserved. The pink ball only lost one, five to four, but the green cube gained eight." But, remember, we're
not conserving velocity. We're conserving momentum. So, momentum is m times v, momentum is mv, and it's a vector, you gotta be careful. It has a direction. So, I'm not saying the amount
of velocity is conserved or anything like that. I'm saying momentum is conserved. So, what I'm saying is
that p initial total has to equal p final total. So, only the pink ball
had momentum initially. So, it had two kilograms times five meters per second of momentum. To start with, the green cube had none. So, this has gotta equal. Afterward, the pink ball has two kilograms times four meters per second. And the green cube does
have momentum afterward. Now, we gotta add it up. There's gotta be the total momentum. So, plus mass of the cube times its eight meters per second. Well, so this isn't too hard. The math here is easy. So, this is gonna be 10 units of momentum, 10 kilogram meters per second is what the ball started with. And then, the ball is gonna end with eight kilogram meters per second. So, we can see right here, the ball lost two units of momentum. That means the green cube better gain two units of momentum. So, it's gonna be plus m times eight. And so, this is what I mean when this can help you solve problems, we can just solve for the mass now. Now, we can know what the
mass of the cube had to be. So, 10 minus eight,
it's gonna be two units. It's how much momentum this
green cube has to gain. And if we divide two by eight, we get that the mass of the green cube had to be .25 kilograms. So, indeed, if we took the
.25 kilograms times the eight that this cube ended with, .25 times eight really does give us two, positive two, it gained two units of momentum. This is why momentum is conserved. Whatever a gain in
momentum one thing gets, there's a corresponding loss in the other. And so, this is gonna be equal. Now, how could you ever make
it so that this was not equal? Well, this will always be equal if the only force is being exerted are internal to your system. The only way you make this nonequal is to have external forces. What would that look like? Well, imagine the ball and
the cube were on a ramp now. So, now that this thing's inclined, gravity is gonna be
exerting an external force. Let's say it sit like 30 degrees. And if we graph the total
momentum of the ball and cube, it's not gonna be a straight line anymore. It is not gonna look like this. That's what it looked like before when there were no external forces. This time, it's gonna be angled up. Let's just call down the ramp positive. This time, it's gonna look like this. Gravity's pulling down the ramp. Let's call that the positive direction. If we're measuring the momentum in that parallel to the ramp direction, we're gonna see an increase
in momentum that direction, because that's the way that this external
force of gravity points. So, how much will this increase? How could you figure out what the change in the total momentum is gonna be? One way to do it is to find the impulse. So, remember, force times time
would give you the impulse. Let's just say we were looking at this for about one second of the time that these were on the ramp. And let's say they have
the values they did before. So, two kilograms and .25 kilograms. That would mean the force down
the ramp is mg sine theta. So, the force along this
parallel direction is m, which the total mass of our
system is 2.25 kilograms. So, m times g, 9.8, times sine of 30. So, you need to take this sine of 30 if you wanna know the change in momentum along this ramp direction. So, the force of gravity
parallel is mg sine theta. And then, that's the force. So, that's the force parallel to the ramp. That would give us the
change in momentum parallel. We just have to multiply by
the time, which is one second. And that gives us about
11 units of momentum. So, this system would
gain 11 units of momentum. That doesn't tell you who's gonna get it. It's not like the two gets all of it, or the .25 gets all of it. But the total system, if you watched for a second, while it's on this
ramp, frictionless ramp, it would increase its momentum. It'd be changing its total momentum because there was an external force. So, recapping, if there are no
external forces on a system, then, the total momentum initial will equal the total momentum final. So, using p equals mv, you can add up these contributions and set them equal to problem-solve. And in a case where there
is an external force, you can find how much that system will gain or lose in momentum by taking that external force times the time that force was applied. That would give you the
total change in momentum of that system over that time.