Distance, Displacement, Speed and Velocity

Describing motion starts with saying how far something has moved. There are two ways to do this, and they are not the same.

Distance is the total length of the path an object travels. It has size but no direction, so it is a scalar quantity.

Displacement is the straight-line distance from the starting point to the finishing point, together with the direction. It has both size and direction, so it is a vector quantity.

Distance is a scalar (magnitude only). Displacement is a vector: the straight-line distance from start to finish, plus a direction.

Worked example — a runner completes one full lap of a 400 m track and returns to the start.

• Distance travelled: 400 m (the whole path).
• Displacement: 0 m (start and finish are the same point).

The two differ whenever the path is not a straight line. If you walk 3 m east then 4 m north, the distance is 7 m but the displacement is 5 m (the straight-line gap), in a north-easterly direction.

Speed tells you how fast an object is moving, without saying anything about direction. Because it has magnitude only, speed is a scalar quantity.

Objects rarely move at a perfectly steady speed. A car speeds up and slows down through a journey, so we often talk about its average speed over the whole trip rather than its speed at one instant.

You are expected to recall some typical everyday speeds. They are estimates, because real values depend on age, fitness, terrain, distance travelled and conditions such as wind:

Motion	Typical speed
Walking	≈ 1.5 m/s
Running	≈ 3 m/s
Cycling	≈ 6 m/s
Sound in air	≈ 330 m/s

These are approximate values. A person's walking or running speed varies with age, fitness and the surface, and the speed of sound changes with air temperature.

Knowing these lets you check whether a calculated answer is sensible. A cyclist travelling at 60 m/s, for instance, should make you suspect an error.

For an object moving at constant speed, the distance travelled is the speed multiplied by the time taken.

$s=vt$

where $s$ is distance in metres (m), $v$ is speed in metres per second (m/s), and $t$ is time in seconds (s).

You must recall and apply this equation. $s=vt$ is not given on the equation sheet.

The equation can be rearranged to find any of the three quantities:

• speed: $v=\frac{s}{t}$
• time: $t=\frac{s}{v}$

Worked example — a cyclist rides at a steady 6 m/s for 40 s. How far do they travel?

$s=vt=6\times 40=240\text{ m}$

The cyclist travels 240 m.

Worked example — how long does sound take to travel 990 m through air at 330 m/s?

$t=\frac{s}{v}=\frac{990}{330}=3\text{ s}$

The sound takes 3 s to travel 990 m.

Velocity is the speed of an object in a stated direction. Because it carries a direction as well as a size, velocity is a vector quantity.

Speed is a scalar. Velocity is a vector: the speed of an object in a given direction.

Two objects can have the same speed but different velocities. A car driving north at 20 m/s and a car driving south at 20 m/s have equal speeds but opposite velocities, because their directions differ.

This distinction matters in physics because forces and accelerations depend on direction. Changing direction, even at a steady speed, means the velocity is changing.

Direction can be given as a compass bearing (north, south-east), as left or right along a line, or as a sign (positive one way, negative the other). Whenever a question mentions direction alongside how fast something moves, it is describing velocity, not just speed.

(Higher Tier only) The idea that circular motion involves changing velocity at constant speed is Higher Tier content.

An object moving in a circle at a steady rate has a constant speed, because the distance it covers each second stays the same. Yet its velocity is constantly changing.

The reason is that velocity includes direction. As the object moves around the circle, the direction it is travelling in changes from moment to moment. Even though the size of the velocity (the speed) is fixed, the direction is not, so the velocity keeps changing.

In circular motion at a steady rate, the speed is constant but the velocity is continually changing because the direction of motion changes.

A car going round a roundabout at a fixed 10 m/s, or a conker whirled on a string, both keep a constant speed while their velocity changes direction all the way round. A changing velocity means the object is accelerating, which is why a resultant force (directed towards the centre) is needed to keep it on the circular path.

1. Treating distance and displacement as the same thing

Distance is the total path length; displacement is the straight-line gap from start to finish with a direction. For a closed loop, distance is large but displacement is zero.

2. Calling velocity a scalar

Velocity is a vector because it includes direction. Only speed is a scalar. If a question gives a direction along with how fast something moves, it is stating velocity.

3. Using inconsistent units in s = vt

Speed must be in m/s and time in seconds to give a distance in metres. Convert minutes to seconds and kilometres to metres before substituting.

4. Quoting typical speeds as exact

Walking at 1.5 m/s, running at 3 m/s and cycling at 6 m/s are estimates that depend on the person and conditions. Present them as approximate, not fixed, values.

5. Saying circular motion has constant velocity

An object moving in a circle at a steady rate has constant speed but continually changing velocity, because its direction keeps changing. That changing velocity is why it is accelerating.