The speed equation: speed, distance and time

The speed equation is speed = distance ÷ time. Rearranged, it gives the distance equation, distance = speed × time, written s = vt, and time = distance ÷ speed. Speed is in metres per second (m/s) when distance is in metres and time in seconds. A runner who covers 400 m in 50 s has an average speed of 400 ÷ 50 = 8 m/s.

The speed equation

Physics uses s for distance and v for speed, which looks odd at first: s is for space covered, and v is the same letter as velocity. Speed has no direction, so it is a scalar; velocity is speed in a given direction, so it is a vector.

The distance equation: s = vt

AQA's GCSE Physics specification gives this form: for an object moving at constant speed, distance travelled = speed × time, s = vt. Students should be able to recall and apply it, and it is also on AQA's equations sheet. For a journey where the speed changes, use the average speed.

Rearranging the speed equation

The speed, distance, time triangle

Some people remember the three forms with a triangle: distance on top, speed and time underneath. Cover the one you want, and the other two show how to work it out: distance over time for speed, distance over speed for time, speed next to time (multiply) for distance. It works because there are only three quantities and two of them multiply to give the third.

Units and converting km/h to m/s

The answer comes out in whatever units you put in: kilometres and hours give km/h, metres and seconds give m/s. To convert between them, remember 1 km = 1000 m and 1 hour = 3600 s:

So 72 km/h is 72 ÷ 3.6 = 20 m/s, and 25 m/s is 25 × 3.6 = 90 km/h. AQA's specification asks students to use ratios and proportional reasoning to convert units like these.

Speed equation worked examples

Example 1: finding speed

A cyclist rides 1500 m in 5 minutes. Find their average speed in m/s.

Convert the time first: 5 minutes = 300 s.

Example 2: finding distance

A train travels at a constant 50 m/s for 3 minutes. How far does it go?

That is 9 km.

Example 3: finding time

How long does it take to walk 1.2 km at 1.5 m/s?

800 s is 13 minutes 20 seconds.

Example 4: average speed over two parts

A bus travels 2 km in 10 minutes, waits, then travels 3 km in 20 minutes. Find its average speed for the moving parts.

Average speed = total distance ÷ total time, not the average of the two speeds:

Example 5: an echo

Someone shouts towards a cliff and hears the echo 2.5 s later. Sound travels at 340 m/s in this example. How far away is the cliff?

In 2.5 s the sound travels 340 × 2.5 = 850 m, but that is there and back, so the cliff is 425 m away.

Here is a quick check.

Distance-time graphs

A distance-time graph shows how far something has gone at each moment. AQA's specification says the speed of an object can be calculated from the gradient of its distance-time graph.

Here x is the time in seconds and y is the distance from the start in metres.

• A straight sloping line means a constant speed. Its gradient is the speed: from 0 to 40 s, 60 m ÷ 40 s = 1.5 m/s.
• A flat line means stationary: from 40 s to 70 s, the walker has stopped.
• A steeper line means faster: from 70 s to 100 s, the gradient is 60 m ÷ 30 s = 2 m/s.
• A curve means the speed is changing. At GCSE Higher, AQA asks students to find the speed at a particular time by drawing a tangent and measuring its gradient.

The average speed for the whole walk is 120 m ÷ 100 s = 1.2 m/s, even though the walker never actually moved at that speed.

Speed and velocity

Speed and velocity use the same equation, but velocity has a direction. Two cars driving at 20 m/s in opposite directions have the same speed but different velocities. A car going round a roundabout at a steady speed has a changing velocity, because its direction keeps changing. AQA asks Higher tier students to explain this for motion in a circle. For velocity with acceleration, see our guide to the SUVAT equations.

Typical speeds

AQA's specification expects students to recall typical speeds for a person walking, running and cycling, typical speeds for different types of transport, and a typical value for the speed of sound in air. It points out that real speeds are rarely constant, and depend on things like age, terrain, fitness and distance travelled. The typical values it gives are:

Common mistakes

• Mixing units. km with seconds, or minutes with m/s, gives a wrong answer. Convert first.
• Averaging the speeds. Average speed is total distance ÷ total time.
• Forgetting there and back in echo questions. Halve the total distance.
• Using the wrong rearrangement. Time is distance ÷ speed, not speed ÷ distance. Check the answer makes sense.
• Reading a flat line as constant speed. On a distance-time graph, flat means stopped.

Practise motion in a short timed paper

GCSE Physics Foundation and GCSE Physics Higher each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so speed and motion turn up regularly but not in every paper. The Laws of Motion course starts with velocity and acceleration graphs.

Read next

Physics · MechanicsDistance-time graphs: speed from the gradient, examplesHow to read distance-time graphs: what flat, straight and curved lines mean, finding speed from the gradient, tangents for accelerating objects, examples.