# Motion and speed

## Key points

• Speed is a measure of how fast an object is moving.

• Calculate speed using the speed equation - speed = distance divided by time.

• The speed equation can be rearranged to find distance travelled and time taken.

## What is speed?

is a measure of how fast an object is moving.

To work out an object’s speed you need to know the it has travelled and the time taken.

Calculate speed using the equation $$speed = \frac {distance}{time}$$.

### The speed equation

Watch this vídeo of Professor Brian Cox explaining the speed equation.

In the video, what units are used to measure speed?

In the vídeo, speed was measured in metres per minute (m/min).

The most commonly used unit in Physics is metres per second (m/s).

Based on what you have learned about the speed equation in the video, have a go at this question.

Calculate the speed of a runner who runs 100m in 10s.

Speed = ?

Distance = 100m

Time = 10s

$$speed = \frac {distance}{time}$$

$$speed = \frac {100}{10}$$

$$speed = 10m/s$$

## Rearranging the speed equation

The speed equation can be rearranged to find either the distance travelled or the time taken.

The speed equation is:

$$speed = \frac {distance}{time}$$

To find the distance the object has travelled, rearrange the speed equation to:

$$Distance = speed \times time$$.

To find the time taken rearrange the speed equation to:

$$Time = \frac {distance}{speed}$$.

Have a look at this example:

Calculate the distance travelled by a car in 10 s, travelling at a speed of 20 m/s

Speed = 20m/s

Distance = ?

Time = 10s

Step 1 - use the speed equation:

$$speed = \frac {distance}{time}$$

Step 2 - substitute in the values you know:

$$20 = \frac {distance}{10}$$

Step 3 - simplify the equation by multiplying both sides by 10 to remove the 10 from the bottom of the fraction on the right hand side:

$$20 \times 10 = \frac {distance \times 10}{10}$$

This cancels to give:

$$200m = distance$$

So the distance travelled is 200 metres

Now have a go at this question.

Calculate the time it takes a runner to cover 200 m at a speed of 8 m/s.

Speed = 8 m/s

Distance = 200m

Time = ?

$$speed = \frac{distance}{time}$$

Substitute in the values you know:

$$8 = \frac{200}{time}$$

To simplify this equation, multiply both sides by time. This will remove 'time’ from the bottom of the fraction:

$$8 \times time = \frac{200 \times time}{time}$$

‘Time’ now cancels from the right hand side, because it appears on the top and bottom of the fraction:

$$8 \times time = 200$$

Simplify this equation by dividing both sides by 8, which will leave time on its own on the left:

$$\frac{8 \times time}{8} = \frac{200}{8}$$

$$time = \frac{200}{8}$$

$$time = 25s$$

So the time taken is 25 seconds.

## Measuring speed

In science, is typically measured in metres per second, m/s. This is the simplest of speed.

is measured in metres.

Sometimes a question will give distance measured in kilometres. You can convert kilometres into metres by multiplying it by 1000.

For example 10 km = 10 x 1000 = 10,000m.

is measured in seconds.

## Speed cameras

Speed cameras are used to find out if a motorist is travelling faster than the speed limit for the road.

The camera takes two photos of the vehicle. The two photos can be taken:

• at a certain time apart so the distance travelled in that time can be measured

or

• a certain distance apart so the time taken can be measured.

The computer connected to the speed camera can then divide the value of ‘distance travelled’ by the value of ‘time taken’ to calculate the speed of the car.

## Relative motion

If you have travelled in a car on the motorway, you may have noticed that other cars passing by appear to move slowly past you, even though you know the actual speeds of the two cars are very high. This is because of their relative motion to each other.

The calculation for relative speed depends on whether the objects are moving in the same direction or the objects are moving in opposite directions towards, or away from, each other.

### Objects moving in the same direction

$$Relative~speed = fastest~speed - slowest~speed$$

For example:

Two cars are travelling in the same direction on a road.

The blue car is travelling at 25 m/s in front of the red car, which is travelling at 30 m/s. What is their relative speed?

$$Relative~speed = fastest~speed - slowest~speed$$

$$Relative~speed = 30 - 25$$

$$Relative~speed = 5 m/s$$

To a passenger in the red car, the blue car appears to be travelling past at 5 m/s.

### Objects moving in opposite directions towards, or away from, each other

$$Relative~speed = speed~of~object~1 + speed~of~object~2$$

For example:

Two cars are travelling towards each other on a road.

The blue car is travelling at 15 m/s. The red can is travelling at 20m/s. What is their relative speed?

$$Relative~speed = speed~of~object~1 + speed~of~object~2$$

$$Relative~speed = 15 + 20$$

$$Relative~speed = 35 m/s$$

To a passenger in either car, it would appear that the other car is approaching at a speed of 35 m/s.