Lesson video

Hello there, I'm Mr. Forbes and welcome to this lesson from the Forces Make Things Change unit.

The lesson's called change in momentum and in it we're going to explore how forces change the momentum of an object when they act over a period of time.

By the end of this lesson, you're going to be able to understand how forces change the momentum of an object when they apply over a period of time.

You are also going to be able to use the equation that links force, mass, change in velocity and change in time to calculate the size of the forces needed to change momentum in a certain amount of time.

These are the key words that will help you understand the lesson.

First is momentum, and that's the product of the mass and the velocity.

The equation is p, for momentum is mv, and that's a vector quantity.

The initial velocity is the velocity of an object at the start of a phase of motion, and we represent that by the symbol u.

And final velocity is the velocity at the end of a phase of motion and we show that as symbol v.

And rate of change of momentum is the change in momentum per second, and that as we'll find out is equal to the resultant force acting on an object.

You can return to this slide at any point in the lesson.

The lesson in three parts and in the first part we're going to be concentrating on calculating changes of momentum, taking into account the direction of movement of objects.

And the second part we'll be linking together force and change in momentum and seeing how the size of a force affects the rate of change of momentum.

And finally, we'll be looking at impact forces in the third part of the lesson and that will allow us to calculate the sizes of forces when we know changes in momentum and time periods.

So let's start by looking at change of momentum.

Let's start by looking at what momentum is again.

So momentum is defined as the mass times the velocity of an object.

In symbols we write that out as p for momentum is m for mass times v for velocity and were again, p is momentum measured in kilogrammes metres per second.

Mass is measured in kilogrammes and that's a symbol m and velocity is measured in metres per second.

The change in momentum of an object is calculated by finding the difference in momentum.

So the change in momentum is just the final momentum of an object minus the initial momentum it had.

We can express that in symbols like this.

We can use the symbol delta p for change in momentum and the initial momentum will be mv minus mu.

And again, the change in momentum delta p measured in kilogramme metres per second, mass of the object is measured in kilogrammes and we've got two velocities here.

Now we've got a final velocity represented by v and an initial velocity represented by u and they're measured in metres per second.

Let's have a look at an example using that equation to find a change in momentum when an object speeds up.

So I've got a cyclist, mass 60 kilogrammes and they're speeding up from four metres per second to seven metres per second travelling in a straight line as shown by the figure here.

And we're gonna calculate the change in momentum.

So what we do is we write out the equation delta p for change in momentum is mv final momentum minus mu initial momentum.

And to make this a bit easier to organise, I'm going to write out the values that I know beneath the diagram.

So my initial velocity u is four metres per second.

My final velocity v is seven metres per second, and my mass is 60 kilogrammes.

So all I need to do now is substitute those values into my equation.

So I get delta p is 60 times seven, so that's 60 times the final velocity there at minus 60 times four.

So that's 60 times the initial velocity for that part, and that gives me a change in momentum so I can calculate the change in momentum and that's 180 kilogramme metres per second.

Okay, I'd like you to do the same procedure here to find the change in momentum for this car.

So I've got a car of mass 1,200 kilogrammes.

You can see it's initial velocity was 1.

5 metres per second, its final velocity, 2.

5 metres per second.

Find the change of momentum please.

So pause the video and find that momentum and then restart.

Welcome back.

Hopefully you selected 1,200 kilogramme metres per second.

We can write down the values again, initial velocity 1.

5, final velocity, 2.

5, mass 1,200, and then write out the expression for change in momentum.

Substitute those values in and that gives us 1,200 kilogramme metres per second.

Well then if you've got that.

We can also find the change in momentum when object slow down.

So let's have a look at example of that.

We've got a ball of mass 9.

15 kilogrammes here and it's going to slow from 10 metres a second to two metres to a second as it rolls across the ground and let's calculate the change in momentum.

So we can write out the equation exactly as we did before and then I write out all the values beneath the diagram to help me organise them.

So I've got initial velocity 10, final velocity 2.

0 and mass 9.

15 kilogrammes.

Substituting those values into the equation and I get a calculated answer of minus 1.

2 kilogramme metres per second.

Well, I get that answer because the momentum afterwards is less than the momentum before.

So I must have momentum being decreased and that's what the minus sign indicates there.

Okay, your turn, and I'd like you to find a change of momentum for this 2,500 kilogramme aeroplane please.

You can see the initial velocity is 20 metres a second and the final velocity 16 metres per second.

So pause the video, find the change of momentum, and then restart please.

Welcome back, hopefully your answer was this, minus 10,000 kilogramme metres per second.

Again, I write up the values to help me organise my calculation.

I write down the equation substitute in the values and that gives me an answer of minus 10,000 per second, which is what you would expect because the aeroplane is slowed down so it has less momentum afterwards.

As we see momentum is a vector quantity and we need to take the direction of movement into account when we're looking at changes in momentum.

So we're gonna look at an example where the direction of movement of the optic changes.

So let's a look at that.

So I've got a ball of mass 9.

5 kilogrammes and it's gonna hit a wall at a speed of four metres per second and it's gonna bounce off the wall in the opposite direction at 3.

9 metres per second.

And I'm asked to find the change in momentum of the ball.

To help me do that, I'm gonna sketch a few diagrams. So first of all, this is my initial condition.

I've got the ball travelling in that direction, 4.

9 metres per second.

And then after it's hit the wall, it's travelling in the opposite direction at 3.

9 metres per second.

So I'm gonna write down the values for all of the quantities I need in my calculation.

My initial velocity is 4.

9 metres per second.

My final velocity is minus 3.

0 metres per second.

'cause now it's travelling in the opposite direction.

It must have an opposite value of velocity.

It's going minus 3.

0 metres per second now.

The masses constant throughout, that's 0.

5 kilogrammes.

So I can now go to my momentum equation, write it out at this, putting the values being very careful with those minus signs.

So you can see my final velocity mv, I've got 0.

5 times minus 3.

9 kilogramme metres per second in there.

So I'm being very careful to take into account the direction.

I calculate the answer to that and that gives me minus 3.

5 kilogramme metres per second.

So it is important that you consider very carefully which direction the object's moving.

One direction will be positive and the opposite direction will be a negative value.

Okay, let's see if you can do a similar calculation.

Again, I've got a ball bouncing off a wall here.

It's got a massive 0.

2 kilogrammes and you can see initially it's going 2.

5 metres per second towards the right and then after it bounces off, it's 1.

5 metres per second towards the left.

So calculate the change in momentum for that ball please.

Welcome back.

Your answer should be minus 0.

8 kilogramme metres per second.

Then I write down all the values.

It shows me that I've got an initial velocity and a final velocity in opposite directions or that minus sign and I do the calculation and it gives me minus 0.

8 kilogramme metres per second.

All done if you've got that.

Now it's time for the first task of the lesson and I've got a group of students talking about changes in momentum when a ball is bouncing off a wall.

I'd like you to explain which of the statements are correct and which are incorrect and correct any mistakes please.

And then I'd like you to calculate the change in momentum of the ship for question two.

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

Welcome back.

Well, Sam and Aisha are correct, momentum is a vector quantity and it depends on the velocity of travel.

As the direction is changing during an impact, the velocity is changing and so the momentum direction must also change.

So I'll have positive and negative momentum there.

Izzy and Alex are both incorrect.

Izzy's not correct because momentum depends on the velocity, not the speed.

Remember, velocity is a vector so that has direction, but speed doesn't.

Alex isn't correct because the ball's moving in a different direction after it bounces, so the momentum must be different even if the speed is the same.

Well done if you've got those.

And now the calculation of momentum for the ship, we've got very large values, but as before, I like to write out each of those quantities quite simply like that.

So initial velocity, final velocity and mass.

Then I can write out the equation and substitute those values in very carefully, taking into account those directions.

And that gives me a change of momentum of minus 7.

7 times 10 to the five kilogramme metres per second.

A very large change of momentum because the mass of the ship is very large.

Well done if you've got that.

Now it's time to move on to the second part of the lesson and we're going to look at the relationship between force time and change in momentum.

We can find the relationship between force change in momentum and time by looking at two equations you may already know.

The first of them is Newton's Second Law, which says results in force is mass times acceleration, which is written in symbols there where F is used to force m for mass and a for acceleration.

And the second equation there is the definition of acceleration, where acceleration is the change in velocity divided by the time, and that's the final velocity minus the initial velocity divided by time.

So we've got symbols like this, a for acceleration, v for final velocity, u for initial velocity and t for time.

If we write out both of those equations, you can see that they've got one thing in common and that's acceleration.

So what I'm going to do is substitute the definition for acceleration into Newton's Second Law because the acceleration in Newton's Second Law is the same as the acceleration in the acceleration definition.

So substituting for a gives this, F equals and then I've got the m from Newton's Second Law.

And then the rest of this equation, v minus u divided it t, that is the definition of acceleration.

So I've now got an expression that's linking force, mass, velocities and time.

The change in velocity v minus u can be represented by the symbol delta v.

So I can write that equation out as this as well.

If I expand out the brackets by multiplying that first version of the equation, then I get this final version.

Force is mv minus mu divided by t.

And that's the equation we most commonly use.

Now as I've got that equation, I can link the rate of change of momentum or how much the momentum changes every second directly to the force.

So if you look, I've got force in this side of the equation and on the other side of the equation this is the change in momentum per second.

It's on the top there, you see the change in momentum, then it's divided by t the time.

So that is the rate of change of momentum.

So I can say the rate of change of momentum of any object is equal to the size of the resultant force acting on it.

So let's see if you understand the relationship between force and change of momentum.

I've got Laura, she's riding off road onto some wet sand.

The force F of the sand pushing on her tyres makes her stop.

How long would it take the sand to stop her if she was travelling twice as fast and had twice the momentum initially? So pause the video, make your decision from the three options there and then restart please.

Well welcome back.

Well it should be two times longer.

The same size force is applying on her, but she's got twice the momentum, so that means it must be two times longer that that force has to work for to bring her to a stop.

Well done if you choose option b.

And here's a second check, very similar to that first one, Laura's riding off the road again into the wet sand.

The force of the sand on her tyres makes her stop.

But how long would it take her to stop if she was travelling at half as fast? So she had half the momentum she had initially.

So pause the video, make your decision and restart please.

Welcome back again, and this time the force would take half as long to bring her to a stop.

She's only got half the momentum, the force is the same, so the time must only be half as long.

Well done if you chose that one.

Now we'll try using the equation in some calculations.

So I'll do one and then you can do one.

What size force is needed to increase the velocity of a 1,500 kilogramme car by six metres per second in a time of five seconds? So as usual, I start by writing out the equation.

Force is mass times change in velocity divided by time.

I can substitute the values based upon the information in the question.

So I've got a massive 1,500 kilogrammes times a change in velocity.

When it started at zero it's increasing to six, so it's 6.

0 metres a second, divide that by the time that's five seconds and that gives me an answer of 1,800 newtons or 1.

8 kilo newtons.

Now it's your go, I'd like you to work out the size of a force needed to decrease the velocity of a 250 kilogramme motorcycle by 12 metres per second in a time of 4.

9 seconds.

So pause the video, work up the size of that force and then restart please.

Welcome back, well your calculation should look very similar to mine with the different values in.

So we've got the force is 250 kilogrammes times, and this is the important part, it's minus 12 metres per second because the velocity is decreased there and divided by the time of 4.

0 seconds.

That's a force of minus 750 newtons.

Well done if you've got that.

We can use the same equation to find other things such as the time it takes for optics to slow.

So we're going to have a look at some examples with that.

We're gonna look at rearranging the equation.

So this is my initial equation, force is mass times change in velocity divide by time.

If I multiply both sides by time I get this and then I can cancel those two t's on the right hand side.

So those two cancel and disappear.

So I end up with Ft equals m delta v and now take get rid of the F, I can divide both sides by F.

So I get this equation and then those two F's cancel on the left hand side and that gives me an equation like this.

The time is the mass times the change in velocity divided by the force.

So I've got a version of the equation I can use to calculate the time it takes for a Force to change the velocity.

Now let's look at examples of using that form of the equation to calculate things.

So I've got a force of 1.

6 kilo Newtons used to increase the velocity of about of mass 2,000 kilogrammes from four metres per second to 12 metres per second.

Calculate the time needed to do this.

So as usual, I write up the initial equation and I can substitute in the values.

So the mass was 2,000 kilogrammes and the change in velocity, well it was at four metres per second and it's gone up to 12 metres per second.

So the change in velocity is 12 metres per second, minus four metres per second.

So I've put that down and I've got a force of 1,600 newtons so I can calculate the time and that's 10 seconds.

Now it's your turn and I'd like you to calculate the time needed to stop this motorcycle.

It's got a massive 400 kilogrammes travelling 15 metres per second and the brakes can produce a force of minus 1,250 newtons.

Pause the video, do your calculation and then restart please.

Welcome back, your answer should look something like this.

And again, it's important to realise you've got a change in velocity from 15 to zero there and you've got a force of minus 1,250 and that gives a total time of 4.

8 seconds for the motorcycle to stop.

Well done if you've got that.

Now it's time for you to use those new skills to calculate some forces times and changes in momentum.

So I've got three questions for you here.

I'd like you to pause the video, read through those questions carefully and answer them and then restart please.

Welcome back and here's the answers to the first question.

So for part a, the increase of momentum is proportional to the size of the force.

If the force is doubled, the momentum is doubled.

So it's now 40 kilogramme metres per second.

In the next situation, the force is halved so the momentum halves.

However, the time is doubled so the momentum gain doubles.

Those two factors cancel each other out.

So we've got the original momentum again, it's the same momentum, 20 kilogramme metres per second.

Well done if you've got those.

And here's the answers to the second part.

Again, I've written out the expressions substituting the values.

So in the first one, I've got a force of minus 2.

8 kilo newtons or 2,800 newtons.

And in the second one, I get a time of 32 seconds for that train to stop.

Well done if you've got those.

Now we've reached the third part of the lesson and we're going to look at impact forces, the forces during collisions between objects.

If there's a resultant force acting on an object, it's going to cause it to change its momentum.

And we can see that by the equation, force is change in momentum divided by the tying the force is acting.

We saw that earlier in the lesson.

F is change in momentum at the top third, mv minus mu divided by time at the bottom the t.

We can write that out in a different format as change in momentum is equal to force times the time the force is acting.

So it looks like this in that situation, mv minus mu equals F times t.

And you should be able to see that the larger the force is, the greater the change in momentum, but also the longer the force acts for, the greater the change in momentum.

So those two factors affect the change in momentum.

Now imagine a golf ball being hit, the momentum of the golf balls going to change because of the force produced by the club on the ball.

The velocity of the golf balls also going to increase or the mass of the golf ball's gonna stay the same.

So the velocity can be increased if we increase the force.

So if you look at increasing the force there, I've got the club and I hit the ball and if I hit it with a larger force, so increase the size of that force, that's gonna result in a greater change in momentum and therefore a greater change in velocity.

I can also increase the velocity further by increasing the time of contact.

So if I've got a golf ball and I hit it for a small period of time, then I've got a small time acting on it.

And so only a small change in momentum.

But if I increase that length of time, then I'm gonna get a greater change in momentum because the club's in contact with the ball producing the force for a longer period of time.

So if I keep that force applying for longer, I get a greater change in momentum.

Okay, I've got a scenario here where I'm hitting a tennis ball.

By following through the red ball is in contact with the tennis racket two times longer than it is with the green ball.

The force F in both situations is the same.

What's the change in momentum of the red ball? Is it two times less than the green ball, the same as the green ball, or two times more than the green ball? So pause the video, make a decision and restart please.

Welcome back.

You should have selected two times greater.

I've got the force acting for twice as long.

It produces twice the change in momentum.

Well done if you've got that.

Let's try that out with some calculations.

So I'm going to do one and then you can have a go.

So again, I've got a golf club, it's hitting a ball with a force of 150 newtons and the impact lasts 0.

01 seconds, just 100th of a second.

The ball's got a massive 0.

5 kilogrammes.

So I'm gonna calculate the increase in velocity of the golf ball.

So start with the expression for momentum here, sorry, change in momentum in terms of force and time.

I rearrange it slightly by dividing both sides by the mass.

So now I've got a change in velocity on the left hand side and on the right hand side I've got force, time and mass.

Substituting all the values I know from the question in, I get this.

So that gives me v minus zero equals 150 newtons times 0.

01 second divided by 0.

05 kilogrammes.

All that's left to do is perform that calculation and it'll give me v, the new velocity of the ball.

And that is 30 metres per second.

Now I'd like you to try that, but this time the contact lasts for 0.

02 seconds because the golfer is following through with the swing, making the club touch the golf ball for longer and therefore giving it a different momentum.

So I'd like you to calculate the increase in velocity of the golf ball.

So pause the video, try and calculate that and restart please.

Welcome back, your calculation should look like this.

You've put all the values into the equation correctly, you should get 60 metres per second and that's what you'd expect.

It's been hit for twice as long, so you'd expect twice the change in velocity for the same force.

Well done if you've got that.

We can do the same soft analysis when an object's being brought to a halt, when its momentum is reduced to zero and its velocity is reduced to zero, we're going to use the same equation as we did before.

But this time, this part of the velocity, sorry, this part of the momentum is being reduced to zero because v is zero.

We can have the initial momentum here.

If that is large, then we can use a large force to stop it or we can use a large period of time or we can use both.

So the size of the force needed to stop something is reduced if we can increase the time of the impact.

Here's an example of that.

A baseball fielder needs to catch a fast ball.

Which of the following will produce the smallest force on their hands as they make the catch? Will it be moving their hands in the opposite direction to the ball, keeping their hands still or moving their hands in the same direction as the ball? Pause the video, make your decision and restart.

Welcome back and it's this one here, moving their hands in the same direction as the ball.

What that will do is increase the amount of time their hands are in contact with the ball to slow it down and that will reduce the force.

Well done if you've got that.

Okay, it's the final task of the lesson now and I've got two questions here.

The first is about a footballer taking a penalty and I want you to explain in terms of momentum, how they could kick the ball so that it will move forward faster once they've kicked it.

And the second one is some calculations of the change in momentum and the size of a force acting during a collision.

So pause the video, work out your answers to those two questions please and then restart.

Welcome back.

Well, in the first scenario there's two ways to make the ball faster.

One is increasing the force of the kick.

So if you kick harder, it will give a larger change in momentum and the ball will be moving faster.

And the second one is keep your foot in contact with the ball for longer for the same force.

And that way the force will act for a longer period of time and produce a greater change in momentum.

And that is a greater velocity change as well.

Well done if you've got those two.

And here are the solutions for the calculation.

First, the change in momentum for the baller.

I've written out the expression substituted in the values and that gives me a change in momentum minus 6.

0 kilogramme metres per second.

And the size of the force during the impact, I use the equation linking forced, time and change in momentum.

So I'll use my value of change in momentum, divide that by the time of the impact, and that gives me a force of minus 30 newtons.

Well done if you've got those.

We've reached the end of the lesson and here's a quick summary of everything.

Momentum is changed when a force acts over a period of time.

The change in momentum is the final momentum minus the initial momentum.

Change in momentum is also the force times the force is acting.

So that gives us an equation, delta p, change of momentum is equal to mv minus mu.

That's the final momentum minus the initial momentum.

And that's also equal to F times t, force times time.

Change in momentum delta p is in kilogramme metres per second.

Mass is in kilogrammes with a symbol m, final velocity is v and initial velocity u, and they're in metres per second and force is in Newtons.

And also time is in seconds and the symbol is t for that.

During any impact, larger forces and larger time periods can be used to produce a greater change in momentum.

Well done for reaching the end of lesson.

I'll see you in the next one.