A cheetah, a snail, a jet plane, you strolling to the shops — everything that moves has a speed. Speed is simply the answer to one question: how much ground do you cover in a certain amount of time? Cover a lot of ground in a short time and you are fast; cover very little in a long time and you are slow.

To turn that into a number we do the obvious thing — we take the distance travelled and share it out over the time it took:

If a bus rolls 100 metres down the road in 10 seconds, then each second it covers 100 \div 10 = 10 metres. We say its speed is 10 metres per second. "Per second" is the giveaway: speed is always an amount of distance for every slice of time.

Units: metres per second and kilometres per hour

Because speed is a distance divided by a time, its unit is always a distance-unit "per" a time-unit. The two you meet most are:

• metres per second (m/s) — the scientist's unit. Handy for people, balls, sound and short bursts. A brisk walk is about 1.5 m/s.
• kilometres per hour (km/h) — the unit on road signs and car dashboards. A car in town might do 50 km/h.

They measure the very same thing — just chopped into different-sized pieces of distance and time. A worked number: our bus at 10 m/s covers 10 m every second, which is 10 \times 3600 = 36{,}000 m — that is 36 km — in one hour (3600 seconds). So 10 m/s and 36 km/h are the same speed.

Swapping between m/s and km/h

That last example hides a lovely shortcut. Turning m/s into km/h always means multiplying by the same number, 3.6, and going back the other way means dividing by it:

Worked example — m/s to km/h. A sprinter runs at 10 m/s. In km/h that is 10 \times 3.6 = 36 km/h — as fast as a car in a quiet street.

Worked example — km/h to m/s. A motorway speed limit is 108 km/h. In m/s that is 108 \div 3.6 = 30 m/s. Now you can compare it directly with the sprinter: the car is three times faster.

Rearranging: finding distance or time

The one formula does three jobs. Cover up the quantity you want and read off what is left — many people picture a formula triangle with distance on top:

Worked example — find the distance. A cyclist rides at 6 m/s for 20 seconds. Distance = speed \times time = 6 \times 20 = 120 metres.

Worked example — find the time. A train needs to cover 150 metres of platform at 30 m/s. Time = \dfrac{distance}{speed} = \dfrac{150}{30} = 5 seconds.

Average speed and instantaneous speed

Real journeys are never at one steady pace. You speed up, slow for a corner, stop at lights. So there are really two different "speeds" worth naming.

• Average speed smooths the whole trip into a single number: \dfrac{\text{total distance}}{\text{total time}}. Walk 100 m to school in 200 seconds and your average speed is 0.5 m/s — even though you dawdled at first and hurried at the end.
• Instantaneous speed is your speed at one exact instant — the number a car's speedometer shows right now. A moment later it may read something else.

A car can average 50 km/h over a trip while its speedometer touches 0 at every red light and 70 on the open road. Same journey, two honest but very different speeds.

See it move

Here is a car on a 100-metre track. Choose a speed and let the time run up: the car slides along by exactly speed \times time metres, and the readout keeps the sum for you. Notice that doubling the speed and doubling the time both move the car further — but a bigger speed eats up the track faster for the same seconds.

How fast is fast?

It helps to carry a few speeds in your head, so a calculated answer can pass a sniff test. If you work out that a person runs at 50 m/s, something has gone wrong — that is faster than a car.

• Walking: about 1.5 m/s (roughly 5 km/h).
• Running hard: about 5 m/s.
• Cycling: about 7 m/s.
• A car on a motorway: about 30 m/s (about 108 km/h).
• Sound through air: about 340 m/s — which is why you see distant lightning before you hear it.
• Light: about 300{,}000{,}000 m/s — the fastest anything can go, nearly a million times faster than sound.