Momentum | Physics | Khan Academy

You have a baseball and a ping-pong ball being hurled at you, let's say at pretty much the same speed. Now, if you try to catch it, you probably know that catching a baseball is going to hurt your hand more compared to the ping pong ball, right? But why? Well, you might say that baseballs are more massive and therefore it hurts more. But why, if something is more massive, would it hurt more? That's an interesting question. And also, because we know that catching a baseball is going to hurt more when you try to catch it, we instinctively pull our hand back, right? That reduces the amount of pain. But again, why? Why does moving your hand back while you're catching the ball reduce the pain?

We're going to try and answer these questions in this video by discovering a brand new quantity called momentum. So let's begin.

So what is momentum? Well, instead of telling you, why don't we just try and figure it out together? Okay, so where do we start? We'll start with Newton's Second Law. You probably already know Newton's Second Law states that the acceleration of an object equals the net force acting on that object divided by its mass. Now let's rearrange this by multiplying by m on both sides. So we will get net force equals mass times acceleration.

All right, what do we do next? Well, let's try and substitute for acceleration. What exactly is acceleration? Remember, acceleration is the rate of change of velocity. So let's write that down: equals m * rate of change of velocity. So that's change in velocity divided by the time taken for the change.

And now I can just put this m over here and write this as change in mv divided by delta t. So let's do this: divide by delta t. Now at this point, you may be like, “Whoa, whoa, whoa! What just happened? Why did you put that m inside? How are we allowed to do that?” And I said that's a great question! Always ask questions whenever a step doesn't make sense to us.

So let's look at it. What really does this mean? If you were to expand that, it basically means mass times change in velocity is basically final velocity minus initial velocity, divided by the time taken for that change. Right? Now I can just multiply this inside, isn't it? So I can do this, right? So this becomes mass times final velocity minus mass times initial velocity, divided by the change in time or the time taken for that change. Right? And that's exactly what this is.

So notice this now represents change in mv, right? So that's what we did over here. Anyways, in doing so, look at what we have achieved. We have now found that the net force acting on a body equals change in this new quantity divided by the time taken for that change. Since we are discovering a new quantity over here, we thought, let's just give it a name, and we call this quantity the momentum. The letter we use for that is p, probably because m is taken for mass, right?

So let's write that down. So net force acting on an object will now become change in momentum divided by the time taken for the change. To try and make sense of what this equation is telling us, let's first investigate what momentum is.

So what we found so far is that momentum, moment, oops, oops! Momentum, momentum, p. What exactly is it? Well, it's this thing: mass times velocity. Okay, so what can we immediately see from here? Well, the first thing we can immediately see is that momentum must be, whatever it is, it's a vector quantity because velocity is a vector quantity. So momentum should have both direction and magnitude.

Another thing we immediately write down is its units. So can you pause the video and see if you can write down its units yourself? Okay, let's see. So the unit of this will be the unit of mass times unit of velocity. Unit of mass is kilogram, standard unit, and the standard unit of velocity would become m/s. So the unit of momentum is kilogram m/s.

Okay, but now let's try to understand what is momentum trying to tell us. Well, we can see that momentum is directly related to mass. That means if you have more mass, you can get more momentum. And you also see that momentum is directly related to velocity. That means more velocity, you get more momentum. But what's important is that it's the product that decides what the momentum is.

So let's take a few examples. Going back to the example of the baseball and the ping pong ball, if they have the same velocity, then which of the two would have more momentum? Can you think about that? Well, let's see. We know that they have the same velocity, so their V is exactly the same. But the baseball has much more mass. The ping pong ball, on the other hand, has a very tiny mass.

And so notice, if you look at the product, the product of the mass and velocity for the baseball is much higher, and therefore the baseball ends up having a higher momentum. Maybe that's got something to do with why it hurts when you try to catch a baseball. But anyways, we'll get back to that.

Just because something has more mass doesn't mean it has more momentum. Well, in this particular case, we knew that they had the same velocity. That's why we could infer that if it has more mass, it should have more momentum, right? But in general, if we don't know about their velocities, then we can't infer anything. We should always calculate the product. It's the product that decides the momentum.

For example, consider a massive train, which has like humongous mass, but at rest. If its velocity is zero, what is its momentum? Well, it's going to be zero. It doesn't matter how massive it is; if it's not moving, it doesn't have any momentum. So a massive train at rest has no momentum. This ping pong ball has more momentum than this train because it's moving. So it's not just the mass; it's the product of mass and the velocity that decides the momentum.

All right, now you might be saying, “Hey, fine! That's great and all, but why should we still care about momentum, right?” Well, for that, let's come back to this equation, and let me just take this and put it over here. Okay! Earlier, Newton's Second Law was saying that force causes an acceleration. But now we can get a look at it from a slightly different perspective. What is this equation trying to tell us?

Well, it's saying that the net force acting on an object equals change in momentum divided by the time taken for the change. In other words, it equals rate of change of momentum. And I love this because this is very similar to how we have seen quantities before. For example, velocity is the rate of change of position. It tells us how quickly position is changing. Acceleration is the rate of change of velocity. It tells us how quickly velocity is changing.

Now we can see net force is rate of change of momentum, which means it tells us how quickly momentum is changing. In other words, if you have a lot of momentum changing per second, that means there must be a huge force acting on you. Net force, the net force acting on you must be a lot. On the other hand, if there's a very small momentum change happening per second, then the net force acting on you must also be very small. A beautiful new perspective of Newton's Second Law!

But another benefit of this formulation is, look, the time over which this force is acting, the time over which that momentum is changing, that's visible to us in this perspective. Well, earlier that was hidden in the acceleration. So it's still there. This is the time over which the force is acting, but it was kind of hidden over here. But now it is, it's there. We can see it, and we will use it later on. We will see how to use it in the next video.

But anyways, let's now apply this to our initial question. Even though these two balls are moving at the same speed, the baseball has a much higher momentum compared to the ping pong ball. The small arrow mark shows smaller momentum over here. Okay! Now, if we stop it, then the momentum goes to zero. So clearly, there is a change in momentum.

Now, let's say if we stop it in pretty much the same time, then we will see that the change in momentum is much larger here because the momentum was bigger to begin with compared to over here, which means we have a much bigger rate of change of momentum over here compared to over here. Since we have a bigger rate of change of momentum over here, that means we're putting a big force on the ball here compared to over here. And if you put a big force on the ball, that means the ball also puts a big force on you, and that's why this hurts more.

And so now you can see why that mass matters because it gives you more momentum. Finally, why is it, while we're catching something heavy like this, we instinctively pull our hand back? How does that help us? How does that reduce the amount of pain that we get? Ooh! When we pull our hand back, we are increasing the time over which that ball is coming to a stop. Right? In other words, we are increasing the time over which that momentum is changing. This number becomes, the denominator becomes bigger. As a result of that, we are deliberately slowing down the momentum change.

So we are reducing the rate of change of momentum. As a result, when you do this, the force acting on you will also reduce. Isn't that beautiful? Almost an everyday application of this new formulation that the net force acting on you equals the rate of change of momentum. In the next video, we'll take this to a whole new level and derive one of the most fundamental principles of physics.