Understanding Conservation of Linear Momentum in Collisions

If an object has a mass m and it's moving with 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. So 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. In 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. We 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. Let'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 momenta. If there are two

particles, it's just P1 plus P2. But if there are more, then you can just add them all up factorially. So 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. Okay? Now, during

this collision, they are 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 are colliding with each other and their individual speeds are changing and their

individual moment are changing, the momentum of the system stays the same even after the collision because there

was no external there is no net external force acting on it. That's what this principle says. Take any system and

doesn't matter how they're interacting with each other. 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. It 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. So 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, v_sub_1,

m_sub_2, and v_sub_2. So, let's clean this up a little bit and think about how do we do that? Well, 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. Let'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 so 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 are moving together with some velocity in some direction. Let's

just call that velocity is we as vi. Then the total final momentum would be the mass times velocity. The total mass

is m1 plus m2 times the velocity v. The 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. Um the hope is that once we get an expression we'll be able to understand the sign and how it

depends upon these variables. Okay. 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. And so 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. They'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.

The first thing I want to think about is the direction of the velocity. Um what decides that? What we see over here is

there's 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. It'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 this if this one's momentum was higher it will end up moving towards the

left. So 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 zero? What would what would happen then? Well, think about it. If m2 is zero, 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 vi1, right? So let's see if our expression gives us that. If m_sub_2

equals 0, then this part goes to zero and this part goes to zero. M1 cancels. Hey, we get v= v1. Perfect. That's

exactly what we would expect, right? Because there is no collision. So we expect it to continue moving with the

same speed. And the same thing would happen if you were to put m1 equal to zero. we would get V equal to just minus

V2 which means you would expect M2 to go towards this direction with just V2. Perfect. 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 what would we

expect once they collide into each other? Think 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. So we would expect this object to just stay put,

stay at rest. In other words, we would expect its velocity to go to zero after collision. And that's exactly what we

would find because look, the numerator goes to zero over here. And so indeed we our equation tells us that it will be at

rest. And from the momentum perspective that makes sense. The total initial momentum is zero because you have plus

mv minus mv m not v not total moment initial momentum is zero. Final momentum should also be zero. So that only way

that can happen if is they stay at rest. Perfect. That makes sense. Is there anything else we can do? Yeah. Um what

if one of their velocities was zero? So let's say v_ub_1 was zero. Okay, let's think about the direction of the new

velocity and its magnitude. Well u if this object were to come and hit it now I know for sure this whole thing should

move towards the left for sure it 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 v_sub_1 is zero, we plug zero over here. This goes out. This goes away. So now look, vi equals negative, right?

Regardless of what values you put for m2, v2, and m1, this will always be negative, telling us that it will be in

this direction. Perfect. It it's it's matching our intuition. That's so awesome. And look at the magnitude. It

is, you know, if you just look at its magnitude, you'll have m2 / m1 plus m2. This is definitely lower than m_sub_2

because this number is smaller than one, the number in front of v2. You see that, right? So, um it matches our intuition

that we should be smaller than the original velocity. So, all of that makes sense. So, yeah, we've done our sense

check. All 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 zero, which is a very ideal situation.

In reality, there will be friction, there will be air resistance and whatnot. So 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? Then 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 zero. They're not exactly zero but the

internal forces are so much higher that we can neglect them. And so in the real world we can apply this to collisions as

long as we are considering initial 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 is it only applicable for collisions? No.

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. Okay, 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 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. But 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. The 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. Since we know that the total change is zero, the sum of the

changes must be zero. So delta P1 plus delta P2 should be zero. Right? That's the whole idea. Or in other words, delta

P1 should equal negative of delta P2. In other words, whatever momentum one particles gains, the other particle

should exactly lose it and vice versa. All right. Now, let's say that the time during which they are, you know,

colliding with each other, the time of collision, let's call as 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. Hey, 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 one. And what is this? This is the rate of change of momentum of

particle 2, which is the force that particle one puts on particle two. That's the only force that's acting on

it. So this is f_sub_1 on two. But what is this saying? This is saying that these two forces must be equal but

opposite to each other. In 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. You can start with Newton's

third law and derive the momentum 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. But the most important insight behind this law came from a German mathematician named Emmy Noithther. 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. It's pretty radical. It's called today as Nother'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 zero the momentum of that

system must remain the same is one of the most powerful principles in physics. Applicable at almost every level but

especially useful for us during collisions or explosions.