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# Displacement, speed and velocity

## Displacement and distance

Distance is the length of the route taken by an object. It is a scalar quantity - it has a magnitude (size) but no direction.

Displacement is a quantity measuring the change in position of an object. Displacement is the vector between the starting position and the end position. Displacement is a vector quantity (i.e. it has a magnitude and direction).

The most commonly used symbol for displacement is $\vecphy{s}$. The symbol $x$ can also be used.

The difference between distance and displacement

A student walks $10 \um$ North, $20 \um$ East and finally $10 \um$ South.

The student travels a distance of $\Tblue{40 \um}$.

The student's displacement is $\Tred{20 \um}$ East.

Distance can never be negative. Displacement can be negative.

## Definition of speed

Speed is the distance travelled per unit of time. It is a scalar quantity.

If you run at 12 km/h, you will cover 12 km in one hour, 6km in half an hour, 24 km in two hours, etc.

Speed is written $S$ or $v$. It is the ratio

$$\Tgreen{\text{speed}} =\frac{\Tred{\text{distance}}}{\Tblue{\text{time}}}.$$
• A car drove 20 km in 20 minutes (1/3 hour). Its speed was $$\frac{\Tred{20\ukm}}{\Tblue{20\umin}} =\frac{\Tred{20\ukm}}{\Tblue{1/3\uh}} = \Tgreen{60\ukmph}.$$
• A plane flew at 1000 km/h for 3 hours. It covered $$\Tgreen{1000\ukmph}\times \Tblue{3\uh} = \Tred{3000\ukm}.$$
• Paul ran 1 km at a speed of 10 km/h. The run took him $$\frac{\Tred{1\ukm}}{\Tgreen{10\ukmph}} = \frac{\Tred{1}}{\Tgreen{10}}\Tblue{\uh} = \frac{\Tred{1}}{\Tgreen{10}} \Tblue{60} \Tblue{\umin} = \Tblue{6\umin}.$$
The ThrustSSC holds the record for the fastest speed ever recorded by a car $(1228 \ukmph)$

## Unit of speed

The SI unit of speed is metres per second $(\text{m/s})$.

Kilometres per hour ($\text{km/h}$) or miles per hour (in the US and UK) are commonly used units in everyday life.

\begin{align*}1\text{ h}&=3600\text{ s} \\1 \text{ km}&=1000\text{m}\\ 1\text{ m/s} = \frac{1/1000 \ukm}{1/3600 \text{ h}}&=3.6 \text{ km/h} \\1\text{ km/h}&\approx 0.28 \text{m/s}\end{align*}

The table shows the approximate speeds of different movements.

m/s km/h
World record for fastest snail 0.003 0.009
Average walking speed 1 3.6
Cruising speed of passenger plane 250 900
Speed of sound in air 340 1,225
Speed of light in vacuum 300,000,000 1,000,000,000
The world record for the fastest snail in a snail race is 0.003 metres per second.

## Average and instantaneous speed

An object's speed at a given moment in time is called its instantaneous speed. It can change over time.

As a car accelerates or decelerates, the needle on the speedometer moves. This is because the instantaneous speed changes. At a constant speed, the needle stays still.

The average speed is the total distance divided by the total duration.

During a particular journey, the car accelerates, reaches 50 km/h, and finally stops. In total, the car travelled 30 km in 45 min. Its average speed is $$\frac{\Tred{30\ukm}}{\Tblue{45\umin}} =\frac{\Tred{30\ukm}}{\Tblue{3/4\uh}} =\frac{\Tred{30}\times \Tblue{4}}{\Tblue{3}} \Tgreen{\ukmph} = \Tgreen{40\ukmph}.$$

If the speed never changes, the object has uniform speed. The instantaneous speed and the average speed are then the same.

Instantaneous speed is the speed measured at a particular instant. The speedometer of a car gives the instantaneous speed.

## Velocity

Velocity is the rate of change of displacement ($\vecphy{s}$) with respect to time.

Velocity is given by the formula: $$\vecphy{v}=\frac{\text{displacement}}{\text{time taken}} = \frac{\vecphy{s}}{t}$$

Velocity and displacement are both vectors. (Remember that speed and distance are both scalars.)

The velocity of an object is always in the same direction as its displacement. Speed and distance have no direction.

Positive and negative numbers indicate movements in opposite directions.

A ship has a velocity of $10 \text{ m/s}$ when moving south and $-10\text{ m/s}$ when moving north. The magnitude of the velocity (and therefore the speed) is the same in both cases.

The velocity of each car changes as it turns along the curved track.