# Displacement, speed and velocity

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.

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.

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 ** 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 |

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.

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.