Lesson video

Hello there.

My name's Mr. Forbes, and welcome to this lesson from the Measuring and Calculating Motion Unit.

In the lesson, we're going to be looking at changes in velocity and using those to calculate acceleration.

By the end of this lesson, you're going to be able to find the change in velocity of an object and use that change in velocity and time measurements to find the acceleration or deceleration of an object.

And here's a set of keywords that help you during the lesson.

The first of them is velocity.

Velocity of an object is its speed in a particular direction.

Second is rate of change, and a rate of change is how much a quantity changes each second, or per second.

Then there's acceleration, which we use to describe the speeding up or change of direction of an object.

And finally, deceleration, which is when an object is slowing down.

And here are some definitions of those keywords that you can return to at any point in the lesson.

This lesson's in three parts, and in the first part, we're going to be looking at changes in velocity.

We're gonna be looking at starting or initial velocities and end or final velocities.

And using those two to find out how much a velocity's changed.

Then we're gonna use that change in velocity in the second part of the lesson to calculate acceleration, where we'll need to know how long it took for that velocity to change.

And in the final part of the lesson, we're gonna be looking at deceleration, which is when an object is slowing down.

So when you're ready, let's start with changes in velocity.

Hopefully, you remember that velocity of an object is the change in displacement each second, and we can express that as a simple equation like this, velocity is changing displacement divided by time.

When we say that velocity is the rate of change of displacement or how much the displacement of the object changes every second, so we can see two examples of that here.

We've got a velocity of five metres per second, and what that means is the displacement is changing by five metres every second.

So after two seconds, the displacement would've changed by 10 metres, or we can have a velocity measured in kilometres an hour like this, two kilometres an hour means the displacement is changing by two kilometres every hour.

So let's practise calculating some velocities.

I've got a question and I'll answer it, and then I'll ask you to do one.

A lorry takes four hours to reach a destination 92 kilometres south of the starting point.

Calculate the velocity of the lorry.

So the first thing we should do is write down the equation we saw earlier: Velocity is displacement divided by time or change in displacement there.

Write that down, velocity is 92 kilometres divided by four hours.

Identifying those two from the question, and finally do the calculation.

It's 23 kilometres per hour, and we've got to give a direction as well.

So it's to the south.

Now it's your turn.

I'd like you to calculate the velocity here.

A jogger runs 280 metres in a straight line in a time of 80 seconds.

Calculate their average velocity.

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

Welcome back.

You should have done your calculation like this, so first of all, write out the equation, velocity is displacement divided by time.

Identify those two values in the question, so the displacement's 280 metres, time was 80 seconds, and we then just do the calculation, 3.

5 metres per second, and we've got to give a direction.

So the only thing we can really put there is along the line of whatever direction they were travelling initially.

Well done if you've got that.

Any object that's speeding up or slowing down has got a velocity change.

And so we've got two values for velocity.

We have the initial velocity, the velocity they started at, and the final velocity, the velocity they finished up at.

And the change in velocity can then be calculated from the difference between those two values.

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

I've got a train travelling east six metres per second, and it speeds up to nine metres per second still east.

What's the change in velocity? Well, all we need to do is to find a change of velocity.

It's final velocity minus initial velocity, and it's very important to write it that way round.

We want to take away the starting velocity from the final velocity.

We substitute in the two values.

So the final velocity was nine metres per second, and the initial velocity was six metres per second.

So just subtract those, and that gives a change in velocity of three metres per second, and the velocity's changing east with direction.

So we should write that down as well.

Because we're dealing with change in velocity, we need to use two symbols.

One to represent the initial or starting velocity, and another to represent the end velocity.

So the initial velocity is represented by the letter U, and the final velocity's represented by the letter V.

We've got to be very careful when using those two symbols because they can look very similar when we write them down.

So I'll be very careful to say which one's which as I go along.

Some examples.

So if I've got a sprinter and they're speeding up from naught metres per second to nine metres per second, u, the initial velocity is nought metres per second, and v, the final velocity, is nine metres per second.

If that sprinter then slowed down from nine metres per second to six metres per second, then u, the initial velocity, is nine metres per second, and v, the final velocity, is six metres per second.

Let's see if we can use those symbols properly.

I'll do one example, and then you do another.

So I've got a skateboarder rolling down a hill in a straight line, they're travelling at two metres per second at the top and 4.

5 metres per second at the bottom.

Calculate the change in velocity of the skater.

So I'll write up my expression, change in velocity is v minus u, making sure I get those two the right way around.

Substitute those values in, v being the final velocity.

It's 4.

5 metres per second, and u being the initial velocity, 2.

0 metres per second.

Do that calculation.

It's 2.

5 metres per second.

Now it's your turn.

I'd like you to calculate change in velocity here, being very careful to use the correct symbols.

A motorcycle speeds up from 3.

2 metres per second to 8.

5 metres per second while travelling in a straight line.

Calculate the change in velocity.

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

Welcome back.

Well, you should have written out its expression, change in velocity is v minus u, final velocity minus initial velocity.

We substitute those two values in, and that gives us a change in velocity of 5.

3 metres per second.

Well done if you've got that.

So, so far we've been describing velocities in terms of north, south, up, down, things like that.

But we can also describe it in terms of positive and negative values.

So we're going to be just using positive and negative values for velocity for most of this lesson.

So for example, if movement to the right is shown by five metres per second, then if we wanna talk about movement to the left, we can just give a value that's negative.

So minus three metres per second would be a movement to the left.

We need to take those positive and negative values carefully into account whenever we're finding changes in velocity.

And that means that we can end up with changes in velocity that are negative as well as positive, and we see some examples of that now.

Okay, let's see if we can do some calculations that involve negative velocities.

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

A stone is thrown into the air at five metres per second, and it's caught moving down at minus three metres per second.

Calculate the change in velocity of the stone.

Well, as usual, I write out expression, change in velocity is v minus u, final velocity minus initial velocity.

And I very carefully put those two values in.

The final velocity was minus three metres per second, and then I take away the initial velocity, which was plus five metres per second.

So I get something like that.

Doing the maths, I get a change in velocity of minus eight metres per second.

Now it's your turn.

A ball is thrown at a wall at six metres per second and bounces straight back at minus five metres per second.

Calculate the change in velocity of the ball.

So pause the video, follow the same procedures I did, and then restart.

Hello again, and so you should have written down the expression, change in velocities, final velocity minus initial velocity, or v minus u.

Substitute those values in very carefully.

Looking at the question, we've got an initial velocity of six metres per second, and a final velocity of minus five metres per second there.

Do the maths, and that gives us minus 11 metres per second.

So the change of velocity is minus 11 metres per second.

Well done if you've got that.

It's time for a quick check to see if you can calculate velocities.

I've got a car driving up a hill at three metres per second.

It stalls and rolls down the hill at four metres per second.

What's the change in velocity of that car if we take up as being the positive direction? So pause the video, work out your answer, and then restart please.

Welcome back.

You should have got a solution of minus seven metres per second.

If we write down the mathematics for that, the change in velocity is minus four metres per second, minus three metres per second, giving a total change of minus seven metres per second.

Well done if you've got that.

Now it's time for the first task.

What I'd like you to do is to complete this table.

The table shows the velocity of three runners as they entered the last section of a race.

It shows the time it took the runners to complete those final 20 metres of the race.

I'd like you to complete that table to show the final velocity, the average velocity over those last 20 metres, and the change in velocity please.

So pause the video, work out all those values, and then restart.

Welcome back.

Your completed table should look something like this.

The math behind each of those calculations is shown here and you can then check how you should have got each of those results.

Well done if you've got them.

Now it's time to move on to the second part of the lesson.

We're going to use those changes in velocity that we've seen to calculate the acceleration of an object.

If the velocity of an object is changing, we say that the object is accelerating, and there are two types of acceleration we can look at.

We can look at acceleration based on changes in speed.

The speed can be increasing or decreasing, and that will cause the object to be accelerating.

So if a car is speeding up, say from six metres per second to nine metres per second, that car can be described as accelerating.

We can also have acceleration if the direction of travel changes.

So acceleration of a car turning a corner at a constant speed is still acceleration.

Even though the speed stays the same, the direction of travel has changed.

In this lesson, we're just going to be looking at objects moving in straight lines that are speeding up and slowing down.

So let's check if you understand what I mean by acceleration.

Which of these statements best describes the acceleration of the trench on in the figure? As you can see, there's a train and it remains stationary at the station.

So is it the train is accelerating to the right, the train is accelerating to the left, the train is not accelerating, or the train might be accelerating or might not be accelerating to the left or right, we just can't work it out from the diagram.

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

Welcome back.

Well, the train's stationary so its speed's not changing and its direction's not changing.

So the train is not accelerating.

Well done if you chose that.

Before we go on to calculating acceleration, it's important to know that acceleration is a vector quantity.

So a vector quantity has a direction associated with it and that direction is very important because if the direction is different, the result is different.

So if I've got an object here and it's stationary, it's at north metres per second and I put an acceleration to the right on it, so the object accelerates to the right, it's going to end up with a velocity towards right, perhaps four metres per second.

So we could describe that acceleration as positive acceleration and because acceleration is a vector, then if we accelerate something to the left, we get a different result, we get an object moving in the opposite direction.

So a velocity of minus four metres per second because we're describing directions as positive and negative.

So it is important to know which direction the object accelerates.

Another check of your understanding of acceleration here.

I've got a car, it speeds up from 20 metres per second to 30 metres per second as shown in that little diagram there.

Which two of these statements are correct? So pause the video, read through the statements, make your selection, and restart.

Welcome back.

You should have chosen these two.

The change in velocity is 10 metres per second.

It's gone up from 20 metres per second to 30 metres per second.

So that's a 10 metres per second increase and the car is accelerating to the right.

It's got faster in the right direction.

Well done if you've got those two.

And here's a more challenging question about acceleration.

Sam throws a ball up, it reaches a maximum height of x.

So if you look in the diagram, you can see the ball and the position shown at equal intervals of time, every 1/10 of a second say.

So it reaches a maximum height of x.

Describe the acceleration of the ball as it rises and only as it rises please.

So pause the video, read through those options, select the correct one, and then restart please.

Welcome back.

You should have chosen the ball is accelerating in a downward direction.

What's happening there is its vertical speed is decreasing so it's slowing down so it must be accelerating downwards.

And if we continued to see what happens to that ball after it's reached high x, we can see it starts moving downwards, getting faster and faster and faster.

So well done if you selected c.

We're going to start calculating accelerations in a minute, but before that we need a definition of what acceleration is mathematically.

An acceleration of an object is defined as the rate of change of velocity and that basically means how much the velocity is changing every second.

So mathematically we express that like this.

The acceleration is the change in velocity divided by the time taken, or in symbols, a equals v minus u divided by t.

So acceleration a is measured in metres per second.

And we'll talk a bit more about that unit in a little while.

Initial velocity and final velocity, so u and v are both measured in metres per second because they're velocities, and time t is measured in seconds, as usual.

I've just said that acceleration is measured in metres per second squared, and that looks like a slightly unusual unit.

So let's quickly try and explain where that unit comes from.

So if we think about velocity, velocity is the rate of change of displacement or how much of displacement changes every second.

And mathematically we've said it's this.

Velocity is change in displacement divided by time and velocity is a distance divided by a time according to that equation.

So to get the unit for velocity, what we do is we get the unit for distance or displacement and divided by the unit by time, and that gives us metres divided by seconds or metres per second.

Acceleration, as we said, is the rate of change of velocity.

So we can do the same thing to try and find the unit for acceleration.

Acceleration is a change in velocity divided by time and as acceleration is a velocity, or metres per second divided by second, we end up with metres per second divided by second, which is metres divided by second squared.

So the unit for acceleration is metres per second per second.

So as we're seeing the expression for velocity now, we can try an example acceleration calculation.

What we're going to do is do that using words and symbols.

So I've got a figure skater and the velocity increases from one metre per second to five metres per second in a time of two seconds.

Calculator her acceleration.

So if we do that in words, we can start by writing out the expression.

Acceleration is change in velocity divided by time and then we substitute in the values, the change in velocity there was five metres per second minus one metre per second.

We're trying to find the difference in velocity and the time is two seconds.

And when we do the calculation, that gives us an acceleration of two metres per second squared.

If we do that in symbols, we write out like this, a equals v minus u.

So that's final velocity minus initial velocity divided by t.

Putting those two values in from the question gives us exactly the same expression, but we're just using the symbol a for acceleration.

And obviously that will give us the same answer, two metres per second squared.

Okay, let's try another couple of examples.

I'll do one and you do one.

And we're going to be using symbols in our calculations here.

So a bicycle speeds up from five metres per second to nine metres per second in two seconds.

Calculate acceleration of the bicycle.

So what I do is I write out the expression equals v minus u over t.

A equals final velocity minus initial initial velocity divided by time.

I substitute those two values carefully from the question, nine metres per second minus five metres per second for v minus u then divided by two seconds and I then calculate the final answer.

It's two metres per second squared.

Now it's your go to calculate an acceleration, a speedboat speeds up from three metres per second to seven metres per second in five seconds, calculate the acceleration of the speedboat.

What I'd like you to do is to follow the same procedure as I did to get the acceleration.

So pause the video and find the acceleration, and restart please.

Welcome back and you should have written this out.

A equals v minus u divided by t.

And then we substitute those two values in, v, the final velocity, is seven metres per second, and the initial velocity, u, is three metres per second divided by the time five seconds.

That gives an acceleration of 0.

8 metres per second squared.

Well done if you've got that.

So far we've looked at fairly simple and small accelerations, but accelerations can be extremely large because changes in velocity can be quite large over very small periods of time.

And we can use standard form to help us with those sorts of calculations.

So for example, during a sporting event, a bullet is fired from a rifle at a target.

The bullet takes 3.

0 milliseconds to reach a velocity of 0.

6 kilometres per second.

Calculate the acceleration.

So to do that we write out the expression that we always use, a equals v minus u divided by t, final velocity minus initial velocity divided by time.

And we substitute those two values in and we can substitute them in standard form.

So v, final velocity, is 0.

6 times 10 to the three metres per second.

That's 0.

6 kilometres per second minus the initial velocity, which is zero.

Then we divide that by the time and the time was three milliseconds, and that can be written out as 3.

0 times 10 to the minus three seconds.

When we put that through our calculator, we can get an acceleration, and that's 2.

0 times 10 to the five metres per second squared.

So a very high acceleration there for a bullet fired from a gun.

So a more challenging question for you here.

I'd like you to calculate acceleration using some standard form.

A particle accelerator can increase the velocity of an electron from nought metres per second to 1.

0 times 10 to the eight metres per second in a time of only 0.

02 seconds.

Calculate the acceleration of the electron and give your answer in standard form.

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

Welcome back.

And the solutions shown here.

We write out the equation as we always do, a equals v minus u divided by t.

And we just put those values in standard form, the final velocity of 1.

0 times 10 to the eight metres per second minus the initial velocity of zero metres per second divided by the time of 0.

02 seconds.

And that gives us an acceleration of 5.

0 times 10 to the nine metres per second squared.

Well done if you've got that.

Now it's time for the second task of the lesson.

And what I'd like you to do is to calculate a range of accelerations for me please.

I've got four different questions there and I just want to know the size of the acceleration in each case.

You don't need to write out the direction for me.

So pause the video, work out the accelerations for those four and then restart please.

Welcome back and let's have a look at the solutions to those.

So the cheetah's acceleration is nine metres per second squared, and you can see the math for that there.

The rocket, 20 metres per second squared.

Well done if you've got those two.

And the second two, the car speeding up, that's an acceleration of two metres per second squared.

And the rollercoaster ride, that's an acceleration of seven metres per second squared.

Well done if you got those two as well.

And now we've reached the final part of the lesson and we're gonna start looking at deceleration and calculating that, and it's when an object is slowing down.

So far we've calculated acceleration for objects that are speeding up, but objects can also slow down.

The velocity can decrease over time.

So for example, if I've got a cyclist travelling at six metres per second and they use their brakes, they can slow down to two metres per second, so the speed's decreased, and that will give a negative value for the acceleration.

And we call those negative values decelerations.

When an object is slowing down, it's decelerating.

Let's see if you understood that with this quick check.

A car slows down from 30 metres per second to 20 metres per second, which two of these statements are correct? So pause the video, read through the statements, select two, and then restart please.

Welcome back.

Well, you should have realised that the change in velocity is minus 10 metres per second.

That indicates that the speed has decreased and that must mean that the car is decelerating.

Well done if you selected those two.

We've already calculated accelerations using this equation and we can use the same equation to calculate deceleration as well.

But we have to be aware that what we'll end up is with negative values.

Okay, it's time to try and calculate some decelerations now.

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

A bicycle brakes and slows down from eight metres per second to two metres per second in four seconds, calculate the acceleration of the bicycle.

So what I need to do is to write out acceleration equation as normal and then substitute in the values, but be very, very careful to identify initial and final velocity.

So my final velocity is two metres per second and my initial velocity is eight metres per second.

And when I do that calculation, I get an acceleration of minus 1.

5 metres per second squared.

And that means you've got a deceleration there.

So now you can have a go.

A sailboat slows from four metres per second to 0.

5 metres per second in seven seconds.

Calculate the acceleration of the sailboat.

Pause the video, do the same calculation style as I did, and find an answer for acceleration and then restart.

Okay, so your stages should look something like this and you should get an acceleration of minus 0.

5 metres per second squared.

Well done if you've got that.

Now we're going to look at more complicated scenarios that involve objects speeding up and slowing down.

So when you've got an object that's moving in one direction and then starts to reverse, the object must have slowed down to reach zero and then sped up in the opposite direction.

So for example, I've got a car and it's moving forward, so I've got a positive velocity written down here of five metres per second.

Then the car would decelerate or accelerate backwards until it stops and reaches nought metres per second.

And if that acceleration continues, what's gonna happen to the car? Well, it's good to start moving backwards in an opposite direction.

So it's gonna end up with a negative velocity.

Throughout that motion of slowing down and speeding up in the opposite direction, the acceleration was always in the same direction.

So that acceleration causes the object to slow down and then speed up in the opposite direction.

When you've got an object that changes direction of travel, you've got to be very careful when calculating any accelerations to take into account the direction of travel.

So I've got a scenario here.

Let's have a look at it.

We've got a ball being rolled up a slope with a velocity of three metres per second, but then it rolls back down again.

After four seconds, it's travelling down the slope with a speed of 2.

0 metres per second.

We're gonna calculate the acceleration of the ball.

So the most important thing is to carefully write out what the initial and final velocities are.

The initial velocity going up the slope u is three metres per second, then the ball is coming back down the slope.

So I'm gonna have a negative final velocity.

So I've got minus 2.

0 metres per second there.

Then I can carefully put those values into my acceleration equation, a equals v minus u over t, being very careful to check those minus signs.

We've got minus 2.

0 metres per second there for the file of velocity.

Then take away 3.

0 metres per second, which was the initial velocity and divide that by the time.

And that will give me a final acceleration of minus 1.

5 metres per second squared.

Let's check if you can do a calculation that involves change of directions.

So I've got a stone, it's thrown vertically upwards into the air at a speed of eight metres per second.

It falls back down to the ground after 1.

6 seconds and it's travelling downwards at a speed of eight metres per second.

Calculate the acceleration of the stone.

So pause the video and see if you can work out the acceleration please.

Okay, welcome back, and let's have a look at the answer to that.

Well, we write out the expression, a equals v minus u over t, and we substitute in the values, being very careful with the direction.

So I've got a final velocity of minus 8.

0 metres per second, and an initial velocity of eight metres per second.

So I subtract those, and that gives me an acceleration of minus 10 metres per second squared.

Well done if you've got that.

Okay, we're at the final part of the lesson now, and I've got a couple of calculations about acceleration for you to do.

So what I'd like you to do is pause the video, read through the questions, do the two calculations, and restart, please.

Welcome back, and let's have a look at the solutions for those two.

And this question shows the importance of reading the information carefully in the question.

So the acceleration when the ball was launched is 12.

5 metres per second squared, and we had to just read those two values for initial and final velocity there from the question and the time, of course.

And for part two, again, looking very carefully at the data in the question, and notice the change of direction, we get an acceleration of minus 56 metres per second squared.

Well done if you've got those two.

Okay, we've reached the end of the lesson now, and here's a summary of everything we should have learned.

Acceleration is the rate of change of velocity, and it's a vector quantity.

Its direction is very important.

Acceleration can be calculated with these equations, acceleration is change in velocity divided by time, and we can write that in symbols as a equals v minus u over t.

Acceleration, a, is measured in metres per second squared; initial velocity, and we use the symbol u for that, and final velocity, the symbol v, are both measured in metres per second, and the time, t, is measured in seconds.

An object that's slowing down can be described as decelerating.

So, well done on reaching the end of the lesson.

I'll see you in the next one.