Conservation of linear momentum

0:00- [Narrator] If an object has a mass, m, and its moving velocity, v, then it has a momentum which is given as the product of the mass times velocity. It's a vector quantity. It has the same direction as that of the velocity. And why do we care about momentum? Because according to Newton's Second Law, the net force acting on an object equals the rate at which its momentum is changing.

0:22So, here's a question. What will happen if the net force on an object is zero? Imagine that this box is moving on a perfectly frictionless surface and there is no air or anything like that. Well, if the net force is zero for some time, then over that time period, the change in momentum must also be zero.

0:45In other words, the momentum of this crate should be a constant. In other words, the crate would continue to move with that constant velocity as long as the net force on it is zero. Now, this is not new. If net force on an object is zero, its velocity stays a constant.

1:02We already know that. But here's where it gets interesting. This idea doesn't just apply to one particle. It can apply to a system of particles as well. Here's what I mean. Imagine instead of one crate, we have two crates of some masses moving towards each other with some velocities.

1:20Let's consider the system of these two particles. How can we extend this idea to a system? Well, now, if the net external force on my system is zero, then it turns out that the momentum of this system, which basically is the sum of individual momentum, if there are two particles, it's just p1 plus p2, but if there are more, then you can just add them all up vectorially.

1:49So the momentum of the system would be a constant. This is called the conservation of linear momentum, and it's one of the cornerstones in physics. Okay, but what is it really trying to tell us? Well, think about it. If these two crates were to come and hit each other and let's say after hitting each other, they will stick to each other.

2:14Okay, now, during this collision, they're putting forces on each other, right? But these are internal forces. The particles are interacting amongst themselves. There is no net external force on any of them, right? So the conservation of linear momentum says even though they're colliding with each other and their individual speeds are changing and their individual momentum are changing, the momentum of the system stays the same even after the collision.

2:44Because there is no external, there is no net external force acting on it. That's what this principle says. Take any system, and it doesn't matter how they're interacting with each other. It doesn't matter how complicated those interactions are, as long as there's no external, there's no net external force acting on that system, the system's momentum cannot change.

3:05It has to stay a constant. And by the way, we call it the conservation of linear momentum because there's another kind of momentum called angular momentum, and there's a conservation principle associated with that as well. But that's not something we're talking about.

3:17So whenever we say momentum, we're talking about linear momentum, okay? But let's try to put this to test. Let's see if we can use this to predict what their velocity is going to be once they stick to each other. Let's try to get an expression for the velocity after the collision in terms of m1, v1, m2, and v2, so let's clean this up a little bit and think about how do we do that?

3:38Well, the whole idea is before the collision and after the collision, the momentum stays the same. Okay, so what's the momentum before the collision? Remember, momentum is a vector quantity. So since we're dealing with one dimensional case, let's just consider, you know, one direction to be positive.

3:52Let's consider the right side to be positive, okay? So what's the total initial momentum of the system? Well, the total initial momentum, which of the system, I for initial, will be the momentum of this object, which is just m1, v1 in the positive direction, plus momentum of this object, which is m2, v2, but the v2 is negative.

4:11So the momentum over here is negative. So minus m2, v2, okay, all right. What's the final momentum of the system? Well, after collision, they're moving together with some velocity in some direction. Let's just call that velocity as v. Then the total final momentum would be the mass times velocity, the total masses, m1 plus m2 times the velocity, v.

4:32The reason we can do that is because they now have moving with the same velocity. So I can treat them as a single object. Notice, I don't even know which direction the velocity is, and therefore I haven't even assigned a sign yet. The hope is that once we get an expression, we'll be able to understand the sign and how it depends upon these variables.

4:50Okay, so according to the conservation of linear momentum, these two momentum must be the same because there was no net external force acting on them. So I can say that the initial momentum should equal the final momentum of the system, therefore this should equal this.

5:08And if I rearrange, boom, I find the final velocity. Now, what I want you to appreciate over here is that during the collision, the forces that we're dealing with could be very complicated. I don't even know how to model them. They might be, and they're definitely varying with time.

5:26They're acting for a very short amount of time. I have no idea about any of that, and yet I can still figure out what the final speed or what the final velocity is going to be just by using this principle. That's what makes it super duper powerful. Now, we might have some questions about this principle, but before we get to those, whenever we derive a new expression, we should see if, you know, we can do some sense check.

5:53The first thing I wanna think about is the direction of the velocity. What decides that? What we see over here is there is a negative sign, okay? So if m1, v1 is larger than m2, v2, then the velocity is going to be positive. Does that make sense? Well, kind of, yes.

6:11It's basically saying the final velocity is in the direction of whichever has a higher momentum, right? So if this one's momentum was higher, then it wins. And so after sticking together, it moves towards the right. On the other hand, if this was higher, if this one's momentum was higher, it would end up going towards the left.

6:28So what matters is not just the mass or the velocity, but the product is what matters. It's the momentum that decides, you know, which direction it'll end up going after sticking. Okay, but let's do some more sense check. What if we substituted m2 to be 0? What would happen then? Well, think about it.

6:48If m2 is 0, we're basically saying that the second box does not exist, okay? Then there wouldn't be any collision at all. So if I ask you what the final velocity is, we would expect it to stay v1, right? So let's see if our expression gives us that. If m2 equals 0, then this part goes to 0 and this part goes to 0, m1 cancels, hey, we yet v equals v1.

7:10Perfect, that's exactly what we would expect, right? Because there is no collision. So we'd expect it to continue moving with the same speed. And the same thing would happen if you were to put m1 equal to 0. We would get v equal to just minus v2, which means you would expect m2 to go towards this direction, which is v2.

7:26Perfect, that makes sense. Okay, what else could we do? Hey, what if both the masses were exactly the same and they were moving with exactly the same velocity towards each other? Then what we would expect, no, don't look at the equation, but what would we expect once they collide into each other?

7:43Think about this. Pause and think about this. Okay, here's how I'm thinking. Since both of them are coming in with the same velocity, same mass, the situation is super symmetrical, right? There is no preference. So there's no reason why this object should, after, you know, sticking together should move towards the right or left because everything is identical.

8:04So we would expect this object to just stay put, stay at rest. In other words, you would expect it'll also to go to 0 after collision. And that's exactly what we would find. Because look, the numerator goes to zero over here. And so, indeed, our equation tells us that it'll be at rest.

8:22And from the momentum point perspective, that makes sense. The total initial momentum is 0. Because you have plus mv0, minus mv0 and not v0, total momentum, initial momentum is 0. Total final momentum should also be 0. So the only way that can happen is they stay at rest.

8:36Perfect, that makes sense. Is there anything else we can do? Yeah, what if one of the velocities was 0? So let's say v1 was 0. Okay, let's think about the direction of the new velocity and its magnitude. Well, if this object were to come and hit it, now I know for sure this whole thing should move towards the left for sure.

9:00It doesn't matter at what speed it comes and hit and what the masses are, right? And I would expect after hitting, the speed would be smaller than before. Let's see if our equation gives us that. If v1 is 0, we plug 0 over here, this goes out, this goes away. So now look, v equals negative, right?

9:19Regardless of what values you put for m2, v2, and m1, this will always be negative, telling us that it'll be in this direction. Perfect, it's matching our intuition. That's so awesome. And look at the magnitude, You know, if you just look at its magnitude, you have m2 divided by m1 plus m2.

9:35This is definitely lower than m2 because this number is smaller than 1. The number in front of v2, you see that, right? So it matches their intuition that we should be smaller than the original velocity. So all of that makes sense. So yeah, we've done our sense check.

9:54All right, now let's get to some of the questions we might be having. First of all, we might ask why do we care? I mean, this principle only works if the net external force on a system is 0, which is a very ideal situation. In reality, there'll be friction, there will be air resistance and whatnot.

10:08So this doesn't apply in the real world, right? And you're right, in general, the friction, air resistance and all the external forces that are unavoidable, they will affect the momentum of the system. But here's the thing. What if we considered the momentum just before the collision and just after the collision?

10:30Then during that collision, the internal forces that these are putting on each other are way higher compared to the external forces like friction or air resistance. And so during the collision, we can assume that the external forces are pretty much 0. They're not exactly 0, but the internal forces are so much higher that we can neglect them.

10:53And so in the real world, we can apply this to collisions as long as you're considering initial case, initial momentum just before the collision and the final momentum just after the collision. So it's applicable, okay? Another question could be, is it only applicable for collisions?

11:10No, this works in general. One more practical example is during explosions, if this object were to, say, explode into two parts, then again, during the explosion, the internal forces are much higher compared to the external ones. So I can use this conservation of linear momentum to get insights about the velocities of those fragments just after the explosion.

11:33Okay, another question is, how do we know this is true? I mean, we just stated it over here. We didn't actually prove it or derive it or something. Well, we have done a ton of experiments, and every single experiment has shown us that the principle is true. It applies at every level from the smallest subatomic particles that you can think of to the largest stars in the galaxies.

11:54But the last question we could be having is do we have any insights behind where does this principle even come from? And yes, we do have a couple of insights. The first one is, let's think about the individual momentum changes. I mean, since the particles are hitting each other, they are putting a force on each other.

12:14The individual momentum does change. It's the momentum of the system that stays the same, but the individual momentum does change. So let's call the change in momentum of this first particle as delta p1 and the change in momentum of the second particle as delta p2.

12:29Since we know that the total change is 0, the sum of the changes must be 0. So delta p1 plus delta p2 should be 0, right? That's the whole idea. Or in other words, delta p1 should equal negative of delta p2. In other words, whatever momentum one particle gains, the other particles should exactly lose it and vice versa, all right?

12:50Now let's say that the time during which they are, you know, colliding with each other, the time of collision, let's call this delta t. Then if I divide this by delta t, this represents the rate of change of momentum of the first particle. And this represents the rate of change, of momentum of the second particle.

13:07Hey, but we know what rate of change of momentum is. That's the net force acting on particle one. In this case, who's putting that force? Hey, that's the force that m2 is putting on m1. So this is the force that particle 2 puts on particle 1. And what is this?

13:24This is the rate of change of momentum of particle 2, which is the force that particle 1 puts on particle 2. That's the only force that's acting on it. So this is f1 on 2. But what is this saying? This is saying that these two forces must be equal but opposite to each other.

13:42In other words, this is Newton's Third Law. When one object puts a force on another, the second object puts back an equal but opposite force on the first one. So what's incredible is that you can derive Newton's Third Law from momentum conservation, or you can do it the other way around.

14:00You can start with Newton's Third Law and derive the momentum of conservation. They're both equivalent to each other. So we can get some more intuition as to why the internal forces does not affect the momentum of the entire system because of Newton's Third Law.

14:12But the most important insight behind this law came from a German mathematician named Emmy Noether. She mathematically proved that all conservation principles, not just that of linear momentum, but conservation of angular momentum, energy, or even charge for that matter, they're all associated with some kind of symmetries in our universe.

14:32It's pretty radical. It's called today as Noether's Theorem. Now you can look it up and it's definitely beyond the scope of our syllabus, but I thought anybody who's learning about this should at least know about her work. Anyways, the long story short, whenever the net external force on a system is 0, the momentum of that system must remain the same.

14:56Is one of the most powerful principles in physics, applicable at almost every level, but especially useful for us during collisions or explosions.