Lesson video

Hello there.

My name's Mr. Forbes.

And welcome to this lesson for the measuring and calculating motion unit.

In the lesson we're gonna look at a range of equations that will allow us to calculate the acceleration of an object, the distance it travels, and the time it takes.

By the end of this lesson, you're going to be able to use two different equations to calculate the average velocity of an object, and you're also going to be able to use an equation of motion that links initial velocity, final velocity, acceleration, distance, and time.

Here's a list of the keywords you'll need to understand for the lesson.

First of them is initial velocity.

And the initial velocity of an object is the velocity it starts up during any phase or part of the motion.

Similar to that, the final velocity of an object is the velocity it finishes that phase of the motion up.

Then we have average velocity.

And the average velocity of an object travelling in a straight line is the change in distance divided by the time taken.

Then we have average acceleration.

And the average acceleration of an object is a change in velocity divided by the time it takes for that change.

And finally, we have uniform acceleration.

And when the acceleration of something is constant, we call that uniform acceleration.

And here's a list of those keywords that you can return to at any point in the lesson if you want to look at their definitions again.

The lesson's in three parts.

And in the first part of the lesson, we're going to look at two different equations that allow us to calculate average velocity.

In the second part of the lesson, we're going to look at a general equation of motion that links together all of the factors involved in motion.

And in the final part of the lesson, we'll look at a series of examples of how we can use different equations to calculate motion.

So when you're ready, we'll begin by looking at the equations for average velocity.

The velocity of an object is not always constant.

It often changes during any journey.

So when you're running a sprint, you'll start off slow and then get faster and faster.

Or when you're in a car, you'll be speeding up and slowing down all the time at different junctions and traffic lights and so on.

Those changes are due to forces acting on the objects and that causes the object to accelerate or decelerate.

So for example, a gravitational force on a falling apple will cause it to accelerate its speeds up as it falls towards the ground.

Or if you're in a car, the braking force will act on it to slow it down and cause it to decelerate.

So in that picture there, we've got car trying to decelerate as it goes around the curve.

Let's check if you understand about acceleration of deceleration.

I've got a large, round stone and it's rolled down a steep slope and has a marker flag positioned on that slope every 10 metres, and you can see them there in the diagram, O, X, Y, and Z.

And I'd like you to do is decide which of those statements are correct.

We've got four statements there, A, B, C, and D.

So pause the video, read through those statements, decide which are correct, and then restart please.

Welcome back.

Hopefully you selected all of them.

All of those statements are correct.

The velocity X is greater than velocity O, and velocity Y is greater than velocity X, and velocity Z is greater than Y, and Z is greater than zero.

The stone's getting faster, as it rolls downhill it's accelerating, so it's velocity is increasing at each stage, O, X, Y and Z.

Well done if you selected them all.

Now we're gonna look at first example of an equation, and this is an equation for the average velocity of an object.

So when an object's travelling in a straight line between two points, we've got an equation for the average velocity and it's equal to the distance between the points divided by the time it takes to move between the points.

So we can write that out as an expression like this.

In symbols, it's much shorter, we can write it out, v equals s divided by t.

And throughout this lesson, we're gonna be mainly using symbols as it's much shorter and quicker to write.

So we'll define those properly.

We've got average velocity, v, measured in metres per second, distance, s, measured in metres, and time, t, measured in seconds.

Okay, it's time for the first check now, and what I'd like you to do is to calculate an average velocity for me.

So I've got a fire engine travelling 9 kilometres north in 15 minutes.

I want you to calculate the average velocity of the fire engine.

And this question involves converting time units and distance units.

So I'd like you to pause the video, work out your answer and then restart, please.

Welcome back.

Hopefully you selected 10 metres per second north, and the way we work that out is.

Well, we can write out the equation, velocity was distance divided by time, or v equals s divided by t.

But we've also got to make sure we convert those distances into metres, so that's 9 times 1,000 metres.

And the time, we need to convert that into seconds, so it's 15 minutes, each of which got 60 seconds, so that gives us a final answer of 10 metres per second.

Well done if you've got that.

In this lesson, we're just going to be looking at objects that accelerate uniformly.

And by uniformly, I mean a constant acceleration.

If we saw that type of acceleration on a velocity time graph, like this, then it would look something like this.

We've got a low uniform acceleration though.

Acceleration is a constant value throughout the whole journey.

We can analyse situations where acceleration changes, but that's much more complex, beyond the scope of this course.

If we've got something moving at uniform acceleration with a higher uniform acceleration, on a graph it would look something like this, a steeper gradient.

If an object's accelerating uniformly a constant acceleration, then we can find the average velocity using the initial velocity and final velocity.

So I've got a line here showing an object with a uniform acceleration, and I can identify its initial velocity, its starting velocity here, it's the starting point of the graph.

And we use the symbol u for initial velocity, and that's 0 metres per second.

We can also find the final velocity, that's the end point of the graph here.

And we use the symbol v for final velocity, and that's 6 metres per second according to this graph.

And to get the average, well, the average velocity is going to be adding those two values together and divided by 2, because there's two values.

So average velocity is u plus v divided by 2.

In this case, the average velocity is 0 metres per second plus 6 metres per second divided by 2, and that gives us an average velocity of 3 metres per second as you probably expected.

And that average velocity can be found even if the initial velocity is not zero.

So the average velocity can be found for any change in velocity as long as the acceleration was uniform, constant acceleration.

So I've got a second graph here and I've got a change in velocity here.

I can identify the initial velocity, u is 4 metres per second, I've read that from the graph there.

I can identify the final velocity, and that's 1 metres per second, reading that off the graph.

And so I can get my average velocity.

Again, average velocity is u plus v divided by 2.

Putting the values that I've read off the graph, find that by two there, and that gives me 2.

5 metres per second.

Okay, I'd like you to use the same technique as I've just used to find an average velocity, please.

So, I'd like to know what's the average velocity between 3 seconds and 6 seconds for the motion shown on this graph.

All of the velocities are in the same direction, so you don't have to worry about velocity, sorry, direction change.

So is it 1.

5 metres per second, 3.

9 metres per second, 3.

5 metres per second, or 5.

5 metres per second? Pause the video, work up the average velocity and then restart, please.

Welcome back.

Hopefully you selected 3.

5 metres per second.

If you look at the graph, you can now identify the two points there.

The initial velocity is 5.

5 metres per second and the final velocity is 1.

5 metres per second.

So I'll write those into the equation, divide 'em by two, and I get a final average velocity of 3.

5 metres per second.

Well done if you've got that.

So, as you've seen, there are two ways of calculating average velocity.

And the one we choose depends upon the information we're given in the question.

If we we're given time and distance, we can use v equals s over t.

If we're given initial velocity and final velocity, we can use average velocity is u plus v over 2.

So I'd like you to decide which of those two equations you'd use to answer each of these questions.

I don't need you to answer them, just select the correct equation.

So, pause the video, read each of the three questions, and then decide which equation you'd use to calculate the average velocity, then restart when you're done.

Welcome back.

Well, for the first one, a rocket travelling 5,000 metres in 10 seconds, you've got a distance and a time there.

So we'd use the equation v equals s divided by t, because we've got distance and time, that could give us the average velocity.

The second one, we've got two different velocities, an initial velocity and a final velocity, so we can find the average velocity there with average velocity is u plus v over 2.

And for the final one, well, we've got two different velocities there.

We can ignore the time, we don't need that at all.

We've just got an initial velocity, 20 metres per second, and a stop, a final velocity of 0 metres per second.

So we'd use the average velocity, as u plus v over 2.

Well done if you selected those three.

Okay, let's try an example of using the equation to actually calculate average velocity now.

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

So I've got a car travelling 520 metres in a straight line, taking 40 seconds, and I wanna work out the average velocity of the car.

What I do is I write out the equation, v equals s over t, because they've got distance and time, fill in the values, and then I can get the acceleration, sorry, the velocity of 13 metres per second.

Okay.

Now, I'd like you to try and calculate the average velocity.

A truck is travelling in a straight line at 8 metres per second, and it accelerates to 14 metres per second.

What's the average velocity of the truck during the acceleration, please? So pause the video, work that out, and then restart.

Welcome back.

You should have selected the other equation.

Average velocity is u plus v over 2.

That's initial velocity plus final velocity divided by 2.

If we substitute the values from the question there, then we can get an average velocity of 11 metres per second.

Well done if you've got that.

Now we've reached the first task of the lesson, and I'd like you to work out some average velocities for me.

And I'd like you to give your answers in metres per second for each of them.

So work out the average velocity for each of those four objects as described there.

Pause the video and restart when you've done that, please.

Welcome back.

Well, let's have a look at the first two.

We've got a car accelerating from 18 metres per second to 22 metres per second along the straight road.

I can use this expression, average velocity is u plus v divided by 2, substitute those two velocities in, and that gives me an average velocity of 20 metres per second.

With a fairground ride launching people up to a height of 56 metres and 4 seconds.

Well, I've got a distance and a time there, so I have to use this calculation using distance and time, and that gives me an average velocity of 40 metres per second.

Well done if you've got those two.

And for the next two, the ship slowing down, well, I've got two velocities there, so the equation I need is average velocities, u plus v over 2, substitute the initial and final velocities in there, and then I've got 3.

7 metres per second.

And for the final one there, well, I've got distance and time, but I've gotta do some conversion.

I've got a time of 3 hours and a distance of 400 kilometres.

So, I can calculate that distance, it's 400,000 metres, and the time is 10,800 seconds, and just use v equals s over t, give me an answer of 37 metres per second.

Well done if you've got that.

In it, we're going to look at an equation of motion, an equation that describes and links distance, time, and velocities.

So let's get started with that.

In many situations, objects move at constant velocity.

They're not speeding up or slowing down.

And they're the simplest scenarios.

And we've already seen, we've got some equations that will allow us to calculate distance and time.

Speed equals distance divided by time, for example.

But there are other times when the object will move with a constant acceleration.

It's getting faster or slower at a constant rate.

It increases or decreases at the same amount each second.

That's called a uniform acceleration, an acceleration that remains constant throughout that part of the motion.

If we combine together equations for acceleration and average velocity, we can come up with an equation that links velocity, distance, and acceleration together.

So let's start by looking at an expression for acceleration.

Acceleration is the rate of change of velocity.

And what that means is, how much the velocity is changing every second.

If we write that as an equation, and you may have seen this before, acceleration is change in velocity divided by time, or a equals v minus u divided by t.

And expressing each of those symbols clearly, acceleration is measured in metres per second squared, initial velocity, u, and final velocity, v, measured in metres per second, and time, t, is measured in seconds.

We get an equation for uniform or constant acceleration like this.

Final velocity squared minus initial velocity squared is two times the acceleration times the distance, or what's shorter written out in symbols, v squared minus u squared equals 2 a s, where acceleration, a, is measured in metres per second squared, initial velocity, u, and final velocity, v, are measured in metres per second, and distance, s, is measured in metres.

And now let's have a look at example of using that equation to calculate motion.

So I've got a question here.

I've got a skier sliding down a steep slope.

He start with a velocity of 0 metres per second and reach a velocity of 10 metres per second in a distance of 25 metres.

And I need to calculate the acceleration of the skier.

So the stages I go through these, I write out the equation, the equation is v squared minus u squared equals 2 a s.

Then I put in the values from the question carefully, and that's probably the most difficult part.

So the final velocity, v, was 10 metres per second squared, put that in, that's 10 squared there.

And the initial velocity, well, it was 0 metres per second, so I'll put that in for the u, and the 2, that's just the number 2, and the the acceleration, which is the thing I'm trying to find out, and finally, the distance, s, 25 metres.

They all go into the equation just like that.

I simplify it by doing all the calculations I can a bit on the left-hand side, that's 10 squared minus 0 squared, that's gonna be 100, then 2 times 25 on the right-hand side, that give me 50 metres times 8, so I've simplified the equation there.

And finally, I need to find a in that expression.

And a is going to be equal to 100 divided by the 50, and that gives me 2 metres per second squared.

The equation can also be used for decelerations, not just acceleration.

So another example here.

I've got a ball rolling along grass pitch.

It starts with a velocity of 6 metres per second, and it stops after it's rolled 36 metres.

Calculate the acceleration of the ball.

The same process again, I'll write out the equation and putting the values carefully from the question.

Looking very carefully at the v and u, the initial velocity and the final velocities, putting those two in there, I've got a final velocity of 0, and an initial velocity of 6.

So those are both squared.

On the left-hand side we've got a distance of 36 metres and we've got that value of 2.

And then simplify by carrying out those squaring and we've got minus 36 is equal to 72 metres times a.

And finally, I can do the calculation to find a, it's minus 36 divided by 72.

That gives me a value of minus 0.

5 metres per second.

And the acceleration is negative there, and that shows me that the ball is slowing down or is decelerating.

Okay, I'll do one more example and then you can have a go.

I've got a cyclist travelling at 5 metres per second along the road.

They see a hazard 10 metres ahead and need to stop before reaching it.

Calculate the acceleration required.

So the process is write out the equation, v squared minus u squared equals 2 a s.

Look carefully at the question, identifying each of those values.

So the initial velocity is.

Sorry, the final velocity is 0, and the initial velocity is 5.

So I've got 0 squared minus 5 squared.

It's 2 times 8 times 10 metres, and that gives me, when I simplify it, minus 25 is 20 metres times a.

And I can then get a by going, minus 25 divided by 20 is equal to a, giving me a final value of a is minus 1.

25 metres per second squared.

So they are decelerating, they are slowing down.

Now it's your turn.

An aeroplane starts at one end of a 1,000 metres long runway, and needs to reach a speed of 40 metres per second to take off at the other end.

Calculate the minimum acceleration required.

So pause video, follow the same process that I've just done, and then try and find that acceleration for me, please, and restart.

Okay, welcome back.

Well, hopefully your calculation looks like this.

We've substituted in the values to the equation that we've written down, gone through each of those stages of simplifying and rearranging, and that gives a final acceleration of 0.

8 metres per second squared.

Well done if you've got that.

The equation can also be used to find a final velocity after a constant acceleration has taken place.

So we'll look at some examples of that.

I've got a hammer and it's dropped from a height of 1.

8 metres, and it accelerates at 10 metres per second squared until it hits the ground.

And the question is, calculate the velocity of the hammer when it reaches the ground.

Well, as before, the stages is just the same, it's just slightly different rearrangements as we go on.

So we write the equation, v squared minus u squared is 2 a s.

We put in the values from the question there, we've not got v, so v squared minus.

Well, the initial velocity was 0, we've got the acceleration, so we put in 10 metres per second squared, and we've got the distance of 1.

8 metres.

So we've put all those values in, and that gives us v squared equals 36.

Now this is just v squared and we've been asked to find v.

So what we've gotta do then is just take the square root of that and that gives us a final value of v equals 6 metres per second.

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

So I've got a feather, it's dropped on the moon this time, from a height of 1.

5 metres and it accelerates at 1.

6 metres per second squared.

And that's because gravity is weaker on the moon.

Calculate its velocity as it reaches the moon's surface.

So we'll go through the same sort of process as before.

Write out the equation, substitute in the values, making sure I use the right acceleration here.

I've got the distance and I've got the initial velocity, and I'm trying to find v.

I find v squared though, it's 4.

8, and that means v is the square root of 4.

8, and so v is 2.

2 metres per second.

Now it's your turn.

A hammer is dropped on Mars from a height of 2.

2 metres and accelerates up 3.

7 metres per second squared.

Calculate its velocity as it reaches Mar's surface.

So go through the same processes I've done, and find its velocity, please.

Welcome back.

Well, hopefully your calculation look something like this.

Again, exactly the same processes, we find v squared, we then take the square root, and we've got our final velocity of 4.

0 metres per second.

Well done if you've got that.

One more example for you to do here.

I've got a rollercoaster, it's got eight 80 metres straight track in it's launch.

It launches carriages from rest, so they're stationary on the start, and it gives them an acceleration of 10 metres per second squared.

Quite a harsh acceleration.

Although the more exciting roller coasters, this one.

Calculate the speed of the carriage at the end of the launch track.

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

And welcome back.

The answer is 40 metres per second.

The calculation you should have carried out looks like this.

We've got the equation, we substitute the values, we get v squared, we take the square root, and that gives us 40 metres per second.

Well done if you've got that.

Okay, now it's time for you to use that equation to solve three questions.

I've got three questions written out here, all of which involve the use of that equation.

I'd like you to pause the video, work out your solutions to each of those showing how you're working, and then restart, please.

Okay, welcome back.

Well, let's have a look at the solutions to each of those.

So for the first one, the stunt driver needs to reach a speed of 20 metres per second from a standing start along a track of length 50 metres to make a jump.

Calculate the acceleration.

Well, we write out the equation, we substitute the values that we can see in the question there to find the acceleration going through each of those stages, and that gives an acceleration of 4.

0 metres per second squared.

For the train, again, we identify.

Well, sorry, we write out the equation, we identify the data we've been given, substitute that into the equation, go through each of the steps of the calculation, and that gives us a distance of 400 metres.

Well done if you've got those.

And for the third question, a rocket accelerating from rest at 2.

0 metres per second squared.

Calculate its speed after it's travelled 40,000 metres.

Again, write up the equation, substitute all of those in, go through each of the stages, so we've got v squared.

Eventually, we can calculate v as 400 metres per second.

Well done if you've got that one.

Now we can move on to the final part of the lesson.

And in it we're going to look at using a range of different equations involving speed, distance, time, and acceleration to solve problems. So let's go on with that.

As you've seen, there are a wide range of equations we use when we're analysing motion, and selecting the right one depends upon the question being asked.

So we need to be able to look at the question, decide on the equation, before we can answer it.

To do that, there are several stages, and these are, we need to identify what quantities being asked for in the question.

We then need to identify what information's being provided in the question.

Then we need to select the correct equation.

Once we've selected that equation, we can substitute values from the question into it, and finally, we can solve the equation.

So let's have a recap of the equations we need to be able to use to calculate motion.

So the first one is velocity is distance divided by time, and we've used that quite a lot in previous lessons.

The next one is the equation for acceleration, and that's the final velocity minus the initial velocity divided by time.

And the equation that we've learned this lesson, v squared minus u squared equals 2 a s, where we've got the final velocity, initial velocity, acceleration, and distance.

Now I've got a question here, I'm gonna select the correct equation to solve it.

So, a sparrow is travelling at an average velocity of 3.

0 metres per second.

How far will it travel if it flies for one hour? All I'm gonna do is identify the equation.

So the first stage is to identify the quantity being asked for.

And in this one, it's how far, so how far is the bird travelling? So that's the distance, s.

The next stage is to identify the information provided in the question to help us solve it.

And that information is shown here.

We've got a velocity and we've got a time.

So we've got.

Sorry, we're asked for the distance, we've been provided with a velocity and a time.

So we need to select the equation that will allow us to find that distance, and that equation is this one, v equals s over t, because all three of those variables are involved in the question.

So I can use that and say s equals v times t, and that'll allow me to calculate s, the distance.

Now I'd like you to select an equation.

So which equation should be used to answer the question in the box there? A cargo transporter is travelling at 4.

0 metres per second.

When it detects rocks ahead, it can decelerate at 0.

1 metres per second squared.

Calculate the distance the boat needs to come to a stop it.

So pause the video, select which equation will allow you to solve that, and then restart, please.

Welcome back.

You should have selected the bottom equation, v squared minus u squared equals 2 a s.

And the reason for that, well, we've been asked for the distance, s, we've been provided with the initial velocity and the final velocity, that's v and u, and we've also been provided with the acceleration.

So we'll be able to answer that question using equation C there.

Well done if you've got that.

Let's have a look at another example.

So, I've got a car travelling at 3.

0 metres per second, and accelerates at 2.

0 metres per second squared for 10 seconds.

Calculate the velocity of the car.

So again, I identify the quantity being asked for, and I've been asked for the new velocity, the final velocity, v, there.

And what information's been provided? Well, I'm being provided with the initial velocity, it was travelling at 3.

0 metres per second, the acceleration, 2.

0 metres per second squared, and the time, it accelerated for 10 seconds.

So I can select the correct equation based upon that.

Got a equals v minus u divided by t, and that will allow me to find the new velocity, v.

Okay, a second one for you to decide on.

I've got three equations again here.

Which of those will allow you to answer this question? At the start of a race, a motorcycle accelerates from a stationary start with a constant acceleration of 3.

0 metres per second squared.

Calculate the velocity of the motorcycle after its travelled 100 metres.

So I'll pause the video, decide which equation to use, and then restart, please.

Welcome back.

You should have selected the bottom equation again here.

V squared minus u squared is 2 a s.

We've been asked to calculate the velocity of the motorcycle, so the final velocity, v, and we've been provided with the initial velocity, it was stationary, the distance it travelled, 100 metres, and the acceleration, which was 3.

0 metres per second squared.

So the equations we used with that one shown, up on C there.

Well done if you've got that.

And one final example of selecting the equation here, I've got a bowling ball and it's rolled across a grass lawn.

The ball is released at a speed of 6.

0 metres per second.

And after travelling a distance of 30 metres, it speeds decreased to 2.

0 metres per second.

Calculate the acceleration acting on the ball as it moves.

So, we identify the quantity being asked for, and we've been asked to calculate the acceleration, a, we identify the information provided, and if you look carefully, we've been given the initial velocity, 6.

0 metres per second, the final velocity, 2.

0 metres per second, and the distance it travelled, 30 metres.

So the equation we need to select is this one, v squared minus u squared equals 2 a s.

And the last quick check for you to select the correct equation.

So which equation is needed to answer this question? The engines of an aeroplane can produce a maximum acceleration of 5.

0 metres per second squared.

How long will it take for it to increase its speed from 140 metres per second to 180 metres per second? So pause the video, select the correct equation, and then restart, please.

Welcome back.

You should have selected a equals v minus u divided by t.

We've been asked to find how long or what length of time it takes, so that's t, and we've been provided with the acceleration, 5.

0 metres per second squared, and the initial and final velocities there.

So well done if you selected that.

Now it's time for you to select the equations and use them to answer a series of questions.

So we've got three questions here, each of them involving the equations that we've learned throughout the lesson.

I'd like you to pause the video and work out the solution showing all you're working, and then restart, please Welcome back.

Well, for the first question, we are calculating acceleration, and your calculation should look something like this, giving you an acceleration of 9.

0 metres per second squared.

And for the second one, we're calculating an average speed, and this is a little bit more complex because we need to actually calculate the velocity in metres per second first, and that's what I've done on the left here.

36,000 metres divided by 3,600 seconds, which is one hour, and that gives us 10 metres per second velocity average speed.

And that will give us a time to travel 200 metres in 20 seconds.

Well done if you got that.

And for the third question, the solution looks like this, we get a distance of 90 metres.

Well done if you've got that one right.

And we're at the end of the lesson.

So here's a summary of everything we've learned.

There's a range of equations we used to describe motion, and they're shown here, v equals s divided by t, a equals v minus u divided by t, and v squared minus u squared equals 2 a s.

And you can see all of the different variables are described there.

Those equations can be used in different combinations to solve a very wide range of questions by identifying the data provided and what's been asked for.

Well done for reaching the end of the lesson.

I'll see you in the next one.

(mouse clicks).