Lesson video

Hello there, I'm Mr. Forbes and welcome to this lesson from the Forces make things change unit.

In this lesson we're going to be looking at conservation of momentum, and momentum is a property of a moving object that allows us to analyze collisions and explosions.

By the end of this lesson, you're going to have an understanding of what we mean by the momentum of an object and be able to calculate momentum.

You're going to be able to use momentum to calculate how objects move before and after explosions or collisions.

Here are the keywords that will help you with the lesson.

And first we have momentum.

Momentum is the product of the mass and velocity of a moving object, and we use the symbol p for momentum.

Then we have the unit of momentum, kilogram meters per second.

Then conservation of momentum is a principle that says that for any closed system, for a collection of objects where there's no external forces, the momentum is always conserved in any interaction within that system.

And then we've got collisions, which is when two objects move together and hit each other, and explosions, where a single object splits or combined objects separate and those pieces of the object move apart.

You can return to this slide later in the lesson if you need to.

The lesson's in three parts, and in the first part we're going to look at what momentum is and how it's calculated, in the second part of the lesson, we'll look at what happens to momentum during collisions of objects, and in the third part, we'll look at what happens to momentum when there's explosions and how we can use it to calculate the motions of objects.

So when you're ready, let's start by looking at what momentum is.

When you have moving objects, some of them are much more difficult to stop than others.

You need a larger force to act on the object to bring it to a stop, or that force needs to act for a longer period of time, and that's because objects have a property we call momentum, the momentum of an object.

I've got two objects here, a large van and a smaller car, and let's imagine they're both traveling at the same velocity of five meters per second there.

The lorry's going to be much harder to stop than the car when they're moving at that velocity.

It's going to need a larger force or a force that acts for a longer period of time.

And that's because the momentum of the lorry is greater than the momentum of the car because it has that greater mass.

So momentum depends upon the mass and the velocity of an object.

The momentum of an object is directly proportional to the mass of the object.

So the more mass of the object is, the greater its momentum, and it's proportional to its velocity as well.

The greater the velocity of the object, the greater the momentum will be.

Let's check if you understand that concept.

I've got two cars here and they've got the same mass, but car X, the top car there in red, is moving at twice the velocity of car Y, the green car at the bottom there.

Which statement is correct? Have a look at the three statements, make your decision, and then restart, please.

And welcome back.

Well, the momentum of car X, the red car, is twice the momentum of car Y, the green car, and that's because it's got a larger velocity and momentum is proportional to the velocity.

So if you double the velocity, you double the momentum.

Well done if you got that.

We define momentum by this equation here, momentum is mass times velocity.

And if we write that in symbols, that p for momentum is m for mass and v for velocity.

Momentum, p, is measured in kilograms meters per second.

That's a fairly unusual sounding unit.

It's just kilograms multiplied by meters per second.

The mass is measured in kilograms and the velocity in meters per second.

Let's try a couple of examples of calculating momentum.

I'll do one and then you can have a go.

I've got a bowling ball of mass 7.

4 kilograms. It's moving with a velocity of six meters per second.

Calculate the momentum of the ball, and I'll write it out in words for this example.

Momentum is mass times velocity.

So I substitute in the values, momentum is 7.

4 kilograms times the velocity of 6.

0 meters per second, and that gives me a momentum of 44 kilogram meters per second.

Now it's your turn.

I'd like you to calculate the momentum of this runner with a mass of 60 kilograms moving at 4.

5 meters per second.

So pause the video, work out the momentum, and then restart, please.

Welcome back.

Well, if you used the symbols, you'd have written p equals m times v.

Substitute the values of 60 kilograms and 4.

5 meters per second, gives us a momentum of 270 kilogram meters per second.

Well done if you got that.

Sometimes we're given the momentum of an object and we need to find its velocity or its mass, depending on what the question asks.

So let's see how we can rearrange the momentum equation to get those.

If we want to find the velocity, v, we can start with the original equation, p equals m times v, momentum is mass times velocity, and divide both sides of that equation by m.

That gives us this relationship.

So you can see I've divided both sides by m.

I'm gonna cancel those two m's on the right-hand side.

So those two there, they cancel each other, m divided by m is one, so we don't need that in the equation any longer.

And that gives me this expression, momentum divided by mass is velocity, and we usually write that in this order, velocity is momentum divided by mass.

So there's my first rearrangement.

Similarly, if I wanna find the mass, m, again, start with the original equation.

This time I'll divide both sides by v, the velocity, so we'll get expression like this.

Cancel the velocities on the right-hand side first, so both of those cancel each other out.

That gives me this equation.

And finally I'll rearrange it, and there I've got an expression, mass is momentum divided velocity.

There's my three versions of the equation.

Let's try and use those versions of the equation in some calculations now.

So I'll try one and you can have a go as well.

A skateboarder of mass 60 kilograms has a momentum of 300 kilogram meters per second.

Calculate the velocity of the skateboarder.

I write out the version of the equation for velocity, velocity is momentum divided by mass, write down the values taken from the question, and that gives me an answer of 5.

0 meters per second.

Now it's your turn to calculate velocity.

We've got all the information here, so please calculate the velocity of that golf ball and then restart the video when you're done.

Okay, welcome back.

The answer should be something like this.

Again, write out the equation, substitute values, and that gives 20 meters per second.

Well done if you got that.

Now let's try calculations of the mass, our third and final version of the equation.

So I've got a radio controlled toy car traveling at five meters per second with a momentum of 15 kilogram meters per second.

Calculate the mass.

So I write out the version of the equation that starts with m equals.

So mass equals momentums divided by velocity.

Write down the values taken from the question, and that gives me an answer of 3.

0 kilograms. And your turn again.

So I'd like you to calculate the mass of the rock based on the information you can see here.

So pause the video, calculate that mass, and then restart, please.

Welcome back.

Well, you should have written an expression like this, and that gives us 15 kilograms. So the mass of the rock was 15 kilograms. Well done if you got that.

One of the important properties of momentum is it's a vector quantity.

It has size, so you can have different sizes of momentum, but the direction is important as well.

So if I have objects traveling in opposite directions, they have opposite signs of momentum.

So let's see an example of that.

I've got objects traveling in opposite directions.

So I've got two cyclists here.

I've got one cyclist traveling towards the right at four meters per second, and they'll have some momentum, and I've got a cyclist traveling towards the left at minus three meters per second.

'Cause the directions are opposite, I've got to state the velocities using positive and negative numbers.

So if I calculate the momentum of the first cyclist, you get an answer like this, 240 kilogram meters per second.

Then I calculate the momentum of the other cyclist, I get an answer like this, minus 210 kilogram meters per second.

So they're in opposite signs.

The momentum direction is important, and we can have positive momentum and negative momentum.

Okay, let's see if you understand why momentum is a vector.

What I'd like you to do is to look very carefully at these six diagrams. All are balls with different masses and different velocities and I need you to identify which pair has exactly the same momentum.

So pause the video, work out which pair has the same momentum, and then restart, please.

Welcome back.

The selections you should have made were C and E.

If you multiply those out, you find that the momentum of each of those is minus 12 kilogram meters per second.

The momentums of the others, well, some of them are 12 kilogram meters per second, you can see that A is 12 kilogram meters per second, but that isn't the same as minus 12 kilogram meters per second.

And similarly, I've got an eight and a minus eight, and they're not the same as each other either.

So well done if you selected C and E.

It's time for the first task of the lesson, and I've got three questions here.

I'd like you to work your way through each of those.

So pause the video, read the questions carefully and answer them, and then restart, please.

Welcome back, and here's the solution to the first couple of questions.

We were asked to calculate the magnitude of the momentum of some objects, and the rock has got a momentum of 2.

4 kilogram meters per second.

The car has got a much higher momentum, it's got a much greater mass and a much greater velocity, so you'd expect that, and that's 45,000 kilogram meters per second.

Then you can rearrange the momentum equation to find a speed or a velocity.

I've started with the original equation, substituted the values for the momentum and the mass, and that gives me 1.

5 meters per second.

Well done if you got all of those.

And the question three here about the drones and they're traveling in different directions.

So we've got drone Y moving upwards and drone Z moving downwards at the same speeds and with the same mass.

They've got different momentum because momentum's a vector property, and so the direction is important.

So they'll have equal amounts of momentum but in opposite directions.

So one will have negative momentum and one will have positive.

So the momentum is actually opposite each other.

Drone Y reverses direction, and we can find the change in momentum where we can find out the initial momentum it had by having its initial velocity and mass, and that gives 6.

0 kilogram meters per second.

And its new momentum, what we do is, well, the direction is opposite, so now it's moving at minus 4.

0 meters per second and that gives us minus 6.

0 kilogram meters per second.

So overall, the change is the difference between those.

It's minus 12 kilogram meters per second.

Well done if you got that one.

Now we're gonna look at why momentum is so important in physics, and to do that we'll look at some collisions and the concept of conservation of momentum.

So let's start that.

Momentum is so useful because of a principle we call the principle of conservation of momentum.

And that is expressed here as the momentum in a closed system is always conserved in any interaction.

You can't increase momentum or decrease momentum in a closed system.

There's always the same amount of momentum no matter what happens.

Now, we need to explain what a closed system is, and a closed system is any collection of objects that are not affected by any external factors or outside things.

So it means that there's no outside forces affecting the system, and there's also no transfer of energy into or out of the system.

No energy is lost or gained by it.

So that's the rules to define a closed system.

But that thing in the box there, the conservation of momentum thing is a very important principle that we use all the time.

We can explain why momentum is conserved based upon Newton's third law.

And if you remember, Newton's third law is this, if object A exerts a force on object B, then object B will exert an equal sized force in the opposite direction on object B.

So forces cause change in momentum, and when one object produces a force that's acting, there will be a change of momentum due to that force, but also, there's going to be a second force which is changing the momentum of another object.

And overall, the change in momentum of one object will cancel out the change in momentum of the other one, so the total amount of momentum will be conserved.

So what I'm saying is that pair of forces cause equal and opposite changes in momentum, so overall there's no change in momentum.

Let's try and explain that in a little more detail using an example and Newton's third law.

So if you imagine I've got two objects approaching each other here, ball A and ball B, and then they collide with each other.

Once they're in contact with each other, object A is gonna cause a force in object B, and that is going to cause a change in momentum of object B.

That force is gonna act on object B, change its velocity, and therefore change its momentum.

But at the same time that change is happening, ball B is going to place a force on object A, and that force is going to be the same size in the opposite direction and it's going to change the momentum of ball A.

So both of those forces are gonna cause changes in momentum, but they're gonna be of different objects.

And overall, the forces are the same size yet in opposite directions and they last for exactly the same amount of time, and that means the momentum of ball A has changed by exactly the same amount as the momentum of ball B, but in an opposite direction.

So momentum must be conserved.

Let's have a look at an example.

I've got about two objects here, X and Y colliding as shown.

The momentum of X changes by three kilogram meters per second.

What's the change in momentum of object Y? So pause the video, make your decision, and restart, please.

Welcome back.

Hopefully you selected minus three kilogram meters per second.

If one object changes by plus three kilogram meters per second, the other must change by minus three kilogram meters per second 'cause the overall amount of momentum hasn't changed.

Well done if you got that one.

We can use the principle of conservation of momentum to analyze things we call collisions, where objects bump into each other.

The stages in these analyses are these.

So first of all, we find the momentum of each object before the collision happens.

That allows us to find the total momentum of the system before the collision.

And what we know by the principle of conservation of momentum is the total momentum after the collision is going to be exactly the same as it was before the collision.

And we use that fact to find any missing momentum, and that will allow us to calculate the velocity or masses of the objects involved.

I'm gonna show you an example of that, because it's much easier to understand once you have an example.

So we've got two objects here.

We've got the first one, object X, it's moving at 2.

0 meters per second and it's got a mass of five kilograms. And I've got object Y here, and that's got a mass of four kilograms and that's not moving before the collision.

So this is my situation before the collision.

After the collision, I've got this.

Object X's velocity has slowed to 0.

4 meters per second and object Y is now moving towards the right there at an unknown velocity.

I'm gonna call that vY.

The masses of both objects are still the same.

Mass doesn't change during the collision.

So my question is to find the velocity, vY, of that second object or stone Y after the collision.

So what I do is look at the momentum before the collision and find out the total momentum of the system, and I do that by looking at the first object.

That's the only moving part in it, but its mass and its velocity, and I calculate the momentum by multiplying those two together and that's 10 kilogram meters per second.

And now, after the collision, I know that the total momentum is still 10 kilogram meters per second.

The momentum of the first stone, the blue one, X there, I can calculate 'cause I've got its mass and its velocity, and that's 2.

0 kilogram meters per second.

And that must mean that the momentum of object Y, stone Y, is the difference in those values.

So there's a total of 10 kilogram meters per second.

Two kilogram meters per second of that is in stone X, and that means that 8.

0 kilogram meters per second is in stone Y.

So the final stage then is to try and find the velocity of stone Y.

And I can do that by taking its momentum, which I know is 8.

0 kilogram meters per second, and dividing it by its mass, which is 4.

0 kilograms. And that gives me the velocity of Y of 2.

0 meters per second.

So it's a multi-stage process, but it allows you to calculate the missing velocity of stone Y there.

Let's have a look at our second example.

I've got two stones sliding and colliding again.

So here's the situation beforehand and here's the situation after the collision.

And again, I'm going to calculate the missing velocity, and this time the velocity of stone X.

So to find that I find first of all the total momentum before the collision, and in this case I've got two objects moving so I've got to add the momentum of both of those two together.

I'm being very careful here and I put brackets around each object's momentum to make it easy to see which mass goes with which velocity.

So you can see there for the first one, I've got my stone at two kilograms and it's moving at 2.

5 meters per second, and I'm gonna add that to the momentum of the second stone.

That's a three kilogram stone and it's moving at minus four meters per second, and that's very important.

I need to take into account that direction.

I work out all of those and get a total momentum of minus 7.

0 kilogram meters per second.

After the collision, the total momentum that I can see in my diagram is just the stone Y there.

So that's got a total momentum of minus three kilogram meters per second, and what that means is I can find the momentum of stone X.

So again, looking at those values and taking into account the direction, I can find that stone X has a total momentum of minus 10 kilogram meters per second.

Using that momentum allows me to calculate its velocity.

Its velocity is its momentum divided by mass, and that gives me a velocity of minus five meters per second.

Okay, I'd like you to try out that process for yourself.

I've got a collision here between two stones again.

You can see the before collision situation and the after collision situation.

And what I'd like you to do is find the velocity of stone Y after the collision.

So pause the video, work out your answer to that, and restart, please.

Welcome back.

Well, the answer to that is six meters per second.

Again, I'll show you the calculation.

The total momentum before the collision is 17 kilogram meters per second.

That must mean the momentum of the stone Y is 12 kilogram meters per second once you've worked out the missing momentum in that second diagram, and that allows you to calculate its velocity as six meters per second by dividing that momentum by its mass, which was two kilograms. Well done if you got that.

We're gonna look at a third example of a collision where we can use the momentum to find out information.

So during collisions, you can have objects that don't bounce off each other, they stick together.

The masses combine to create a new object.

So I've got an arrow and it's got a mass of 0.

3 kilograms. It's traveling at 20 meters per second.

And let's imagine that hits a coconut that's got a mass of two kilograms and they stick together and they move off together.

So a situation like this beforehand.

Before the collision, I can work out the total momentum.

That's just the momentum of the arrow, and that's 6.

0 kilogram meters per second, multiplying the mass by the velocity.

After the collision, the momentum will be the same, it's still 6.

0 kilogram meters per second.

So I've got this object where the arrow and the coconut combined together, but I still know the total momentum.

So what I can do is find the total mass of the two objects joined together.

I know the momentum, so I divide the momentum by the total mass, and that gives me a velocity of 2.

61 meters per second.

So the arrow is slowed down considerably and the coconut has started moving as they're joined together.

Okay, it's time for you to try some momentum and collision calculations, and I've got two questions here for you.

So what I'd like to do is pause the video, read them carefully, work out the answers, and then restart, please.

Welcome back, and here's the solution to the first of those questions.

We find the momentum before the collision of 8,000 kilogram meters per second.

Then we find the combined mass of the two objects, 20,000 kilograms. So we divide that initial momentum by that mass and that shows that the objects move off at 0.

4 meters per second.

Well done if you got that.

And the second question's here.

We find the total momentum before the collision, 0.

48 kilogram meters per second.

Then after the collision, we can find out the momentum of ball W, the white ball there.

It's minus 0.

16 kilogram meters per second.

Remember, we take the direction into account.

That allows us to find the momentum of the yellow ball, ball Y.

That momentum is 0.

64 kilometers per second.

Again, we're very careful with the directions there.

And finally, we can use the momentum of the yellow ball and its mass to find its velocity, which is 4.

0 meters per second.

Well done if you got that.

Now it's time to move on to the next part of the lesson, and in it we're going to look at principle of conservation of momentum and how it applies to explosions, situations where objects move apart from each other.

So let's start that.

Momentum is conserved in any situation in a closed system.

So it's not only conserved when objects collide with each other, it's conserved when object break apart, and when objects break apart, we call those situations explosions.

So let's have a look at an example.

I've got an object here and it's made up of two parts, part X and part Y.

And let's say that this object isn't moving initially, so it's got an initial momentum of zero because no matter what its mass was, because it's got a velocity of zero, if you multiply any mass by a velocity of zero that's gonna give a momentum of zero.

So I've got a momentum of zero kilogram meters per second.

Now imagine that this object breaks apart, there's springs or something between the two sections and they're released and the object flies apart at different velocities, like that.

The two objects are moving in opposite directions.

We call that event the explosion.

The momentum after that event is also going to be zero.

There was no momentum before the explosion, so after the explosion, the total momentum of the system is going to be zero again.

We can investigate momentum in explosions using dynamics trolleys.

The dynamics trolleys are those small wheeled vehicles you use in motion experiments.

So imagine I've got two trolleys here next to each other, and initially they're stationary, so the total momentum of this system is zero.

Now, the trolleys can have a little springs between them and you can trigger those springs to push the trolleys apart to make the explosion.

If we release it, the trolleys are going to be pushed apart and the momentum is gonna be something like this.

I've got trolley A moving off to the left here, and that's gonna have a momentum, and we call that momentum A or pA, and the other trolley is gonna have a momentum pB.

Now, what we can say is, well, the total momentum is still zero after that explosion.

What that's gonna mean is the trolleys have got equal and opposite momentum.

For the momentum to still be zero, pA and pB added together must equal zero or pA is equal to minus pB.

Let's have a look at an example of using the conservation of momentum to find out the velocity of an object after an explosion.

So I've got two dynamics trolley here.

Before the explosion, before I release the spring, they're not moving, they're at rest.

And I've got the masses of the two trolleys written down there.

After the explosion, you can see the trolleys are both moving.

Trolley A is moving to the left at 0.

5 meters per second, and trolley B is moving to the right at an unknown velocity.

So the first thing I can say is I'm gonna find the velocity of trolley B after the explosion.

Before the explosion, the total momentum must be zero because the trolleys were at rest.

So I've got a total momentum of zero.

After the explosion I know that the total momentum must be zero, so if you add the momentum of the two trolleys, pA and pB, that's equal to zero kilogram meters per second there.

So pB is minus pA.

So I then write out pB equals, and I work out the momentum of trolley A.

It's 2.

0 kilograms times 0.

5 meters per second and it's minus that for trolley B, and that gives me a momentum of trolley B of minus 1.

0 kilogram meters per second.

Now I can find the velocity of trolley B.

It's the momentum divided by the mass.

I've got the momentum, it's minus 1.

0, divided by the mass, 0.

8 kilograms, and that gives me a velocity for trolley B of minus 1.

25 meters per second.

And it's minus because it's moving in the opposite direction to trolley A.

Okay, I'd like you to try an example of that now.

I've got two stationary trolleys again.

They're both at rest.

The momentum of trolley X after the explosion is four kilogram meters per second.

What must the momentum of trolley Y be? So pause the video, work out your answer, and restart, please.

Welcome back.

Well, it must be answer C, minus four kilogram meters per second, because they must gain equal and opposite momentum during the explosion.

Well done if you got that.

Okay, now it's time for the final task of the lesson.

I've got two questions here.

The first of them I'd like you to complete those statements, and in the second one what I want you to do is go through the process of calculating momentum and I want you to find the velocity of trolley B after an explosion.

So pause the video, work out your answers, and restart, please.

Welcome back.

Here's the answer to the first one.

So if two joined objects are at rest, the total momentum is zero kilogram meters per second, and the momentum is zero kilogram meters per second when they break apart in an explosion because momentum is always conserved.

Well done if you got those.

And here's the calculation for the second part.

Again, we find the momentum before the explosion.

It's zero.

We know that the momentums are opposite to each other, so pB is minus pA.

So we can find pA and just put a negative sign in front of it and that gives us the momentum of pB as minus 1.

8 kilogram meters per second.

And finally, we can use that momentum and the mass of that trolley to get a velocity for B of minus 0.

9 meters per second.

Well done if you got that.

We've reached the end of the lesson now, and here's everything we've learnt.

The momentum of an object is given by momentum is mass times velocity, or in symbols, p for momentum is m for mass times v for velocity.

Momentum is measured in kilogram meters per second, and it's a vector.

It has direction and that direction is very important.

The principle of conservation momentum says: the momentum in a closed system is always conserved in any interaction, and that's always true.

This includes collisions and explosions, and we can analyze those by remembering that the total momentum before an event is equal to the total momentum after an event.

Well done for reaching the end of the lesson.

I'll see you in the next one.