Physics
Cambridge IGCSE

TOPIC 1A: FORCES and MOTION

# 1.2a Speed and acceleration

To begin this section, we need to know how to measure motion: Some of this you should have covered already in previous learning, but measuring simple motion is still included in the IGCSE syllabus.

## Speed

Do you know how fast you can run? Do you know how you would measure your fastest running speed?

To do this, you need to know two key measurements - the distance you have travelled, and the time it took you to cover this distance.

The fastest runners in the world can run 100 metres in just 10 seconds. That means they cover 10 metres each second, or 10 metres per second. This is written as 10 m/s.

Figure 2. Measuring your running speed.

To calculate the speed, we just need to divide the distance travelled at some point by the time it took to cover that distance:

 speed (m/s) = distance moved (m) time taken (s)
 v = s t
Learn this formula!

What if the runners' speed changes during the motion? In this case we need to calculate the average speed. To do this we divide the total distance travelled by the total time it took:

 average speed (m/s) = total distance moved (m) total time taken (s)
 v = s t
Learn this formula!

Why is it the average speed? It might be that the runner was faster towards the end of the race after a slower start. So the answer is actually the average speed over the race, not the speed at any point. Sometimes the examiners will say 'average speed' and sometime just 'speed', assuming it stays constant. These formulas are essentially the same, you can just learn one of them!

Note: Take care - the letter used for speed in this syllabus is 'v' and for distance is 's'. You may have learned different letters - you need to start using these ones to get used to the formula!

Example:

The runner in figure 2 completes 400 m is a time of 160 seconds. What is her average speed?

We know that:

 average speed = total distance moved = 400 time taken 160
so the average speed = 2.5 m/s

Questions:

1. A feather falls for 8 seconds at a speed of 0.4 m/s. How far has it fallen?

We know that:
 average speed = distance moved time taken

Rearranging this to find the distance travelled gives:

distance = speed x time = 0.4 x 8

so the distance travelled = 3.2 m

2. An athlete completes a 4 km mountain race in at an average speed of 3 m/s. How long does the race take?

The distance is given in 'km' and needs to be converted to metres:
4 km = 4000 m.
We now need to rearrange the formula to have the time on the left of the formula:
 time taken = distance moved = 4000 average speed 3

so the time taken = 1333 seconds. ( We could round this up to 1330 s, to 3 sig figs.)

### Speed and velocity

In Physics, the speed of an object is often referred to as the velocity, also measured in m/s. Speed and velocity are (almost) the same - velocity is just the speed in a given direction. Velocity can have a positive value for an object moving forwards, and also a negative value when the object is moving backwards.

## Acceleration

When we speed up - increasing our velocity - we are accelerating. When we slow down we are decelerating.
Acceleration is a measure of how quickly the speed of an object is changing, and is measured in 'metres per second squared', or m/s2. A high acceleration means the speed changes rapidly. A negative value for acceleration means the object is decelerating.

One of the most common examples of acceleration in physics is when an object is dropped, and this is called 'free fall'. If we drop any object, gravity makes it accelerate downwards. This initial acceleration is always about 9.8 m/s2 on Earth (assuming air resistance is negligible). This value is given the letter 'g' for gravity:

• The acceleration of gravity on Earth (g) = 9.8 m/s2   (near to the surface of the Earth)

This gives us an easy way to find the value of g - by measuring the acceleration of an object falling under gravity. It is best to use a dense compact object to reduce the effect of air resistance to get a reasonably accurate answer.

## Measuring Acceleration

Imagine travelling to school on a bus, or in a fast car:

Fig 3 . A school Bus and Car

Which of these accelerates the fastest at the beginning of the journey?

Both can accelerate from zero to 10 m/s - a safe driving velocity around town. However, the car will get to this velocity in much less time. To find the acceleration of the vehicles in figure 3, we need to know the change in the velocity, and also the time taken to do this.

If the car takes five seconds to reach 10 m/s, then the velocity has changed by 2 m/s each second. (2 m/s per second). We write this as 2 m/s2, and this measurement is the acceleration of the car. The units are "metres per second squared".

### Notation

There are some standard symbols used in this topic that you need to learn:

• Δv ('delta' v)= the change in velocity (m/s)
• t = the time taken (s)
• a = acceleration (m/s2)

Here is the formula used to calculate acceleration:

 acceleration (m/s2)= change in velocity (m/s) time taken (s)

 a = Δv t

Learn this formula!

Example:

The car in figure 3 accelerates from rest to a velocity of 30 m/s in a time of 8.5 seconds. Calculate the acceleration of the car.

We know that it started at rest, so the change in velocity Δv = 30 m/s.

 acceleration = change in velocity = 30 time taken 8.5
so the acceleration = 3.53 m/s2

Questions:

3. The school bus in figure 3 accelerates from rest to a velocity of 16 m/s in a time of 20 seconds. Calculate the acceleration of the bus.

We know that the change in velocity (Δv) is 16 m/s:
 a = Δv = 16 t 20
so the acceleration = 0.8 m/s2

4. A cheetah is widely believed to be the fastest animal on Earth. It can accelerate at 2.5 m/s2. If the cheetah starts at a velocity of 3 m/s, how long will it take to reach a velocity of 13 m/s?

In this question, the change in velocity Δv = 13-3 = 10 m/s.

We now need to rearrange the formula to have the time on the left of the formula:
 t = Δv = 10 a 2.5
so the time taken = 4 s.

We know that:
 average speed = distance moved = s time taken t

However if an body is accelerating from an initial velocity u to a final velocity v, then we need to find the average velocity (speed). To find the average of any 2 numbers, we add them together then divide by 2:

 average speed = v+u 2

Therefore we can put these two formulas together to give:

 v+u = s 2 t

1. Can you rearrange this last formula to put 't' as the subject?
2. Can you make 't' the subject for the formula for acceleration?
3. Can you combine these 2 to get the final formula given in the blue box above?

### Standard Practical: Investigating the motion of everyday objects such as toy cars or tennis balls

As stated clearly in the title here, you should be able to use basic measuring skills to record the motion (typically speed or acceleration) of simple objects.

An example of this would be to roll a toy car or ball down a very gentle long slope as shown in the diagram.

Some examples of experiments you could do are:

1. Finding the average speed:

Measure the distance from A to D, then time how long it takes the car to roll from A to D. Find the average speed by using the formula for speed shown above.

You could also find the speed over any of the shorter sections marked above, and compare as it rolls down the slope.

2. Finding the acceleration:

• Measure out distances 20cm apart (A to B, and C to D on the diagram shown above).
• Time how long it takes to travel A to B and also the time from C to D. Use this data to find the speed at the top of the slope, and also the speed at the bottom.
• Now time how long it takes to travel from the centre of the top section (AB) to the centre of the bottom section (CD). Alternatively measure the distance between these centre points.
• Use the results to calculate acceleration from either of the acceleration formulas above :

If we know the change in velocity (Δv) and time taken (t) we can use a = Δv /t

Note: Experiment 2 is much easier if we use light gates or similar technology to work out the velocity of the car for us.

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