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How fast are things?

Change and motion · level 4, lesson 6 of 6

A question to start

Put these in order, slowest to fastest:

  • a cheetah sprinting,
  • the Curiosity rover driving on Mars,
  • the sound of a clap travelling across a room,
  • a person walking,
  • a car on a highway at 100 km/h.

Easy? Here is the catch. The facts come in different units. The rover's top speed is about 4 centimetres per second. Sound travels about 343 metres per second. The car does 100 kilometres per hour. You cannot compare 4, 343 and 100 when they are counting different things. One of the surprises waiting for you: the cheetah and the highway car are almost exactly as fast as each other.

The idea

Same units first

A speed is a rate: a distance per a time. Two speeds can only be compared fairly when they use the same distance unit and the same time unit. So the first job is always to convert: change one speed into the other's units.

The two most common speed units are:

  • metres per second (m/s): good for people, animals, sound;
  • kilometres per hour (km/h): good for cars, bikes, trains.

Why 1 m/s is 3.6 km/h

Something moving at 1 m/s goes 1 metre every second. How far in an hour?

  • One hour is 60 minutes, and each minute is 60 seconds: \(60 \times 60 = 3600\) seconds.
  • So in an hour it goes 3600 metres.
  • 1000 metres make a kilometre, so 3600 metres is \(3600 \div 1000 = 3.6\) km.

So 1 m/s = 3.6 km/h. That gives two rules:

  • m/s to km/h: multiply by 3.6. (Each m/s is worth 3.6 km/h.)
  • km/h to m/s: divide by 3.6.

A check that you used the right one: the km/h number is always the bigger number, because a kilometre is a big unit but an hour is a long time, and the long time wins. 10 m/s is 36 km/h, not 2.8 km/h.

Other units

You can convert to anything by the same thinking: ask "how far in one of the new time units?", then change the distance unit.

  • cm/s to m/s: 100 cm make a metre, so divide by 100. 4 cm/s is 0.04 m/s.
  • cm/s to km/h: 4 cm/s is 0.04 m/s, and \(0.04 \times 3.6 = 0.144\) km/h.
  • km/s to km/h: an hour has 3600 seconds, so multiply by 3600.

A table of speeds

Here are typical speeds, all in m/s and in km/h. Some are measured exactly; others (like the cheetah) vary from animal to animal, so they are rough.

Thing m/s km/h
Human hair growing (roughly 1 cm a month) about 0.000 000 004 about 0.000 000 014
Curiosity rover on Mars, top speed 0.04 0.144
Person walking about 1.4 about 5
Usain Bolt's 100 m world record (average) about 10.4 about 37.6
Car on a highway about 27.8 100
Cheetah sprinting (short bursts) roughly 28 roughly 100
Passenger jet cruising about 250 about 900
Sound in air at 20 °C about 343 about 1235
Earth moving round the Sun about 30 000 about 107 000
Light about 300 000 000 about 1 080 000 000

Some of these deserve a closer look.

  • Usain Bolt ran 100 m in 9.58 seconds in 2009. His average speed was \(100 \div 9.58\), about 10.44 m/s. His top speed during the race was higher, because he started from standing still.
  • The cheetah really is about as fast as a car on the highway, but only for a short sprint of a few hundred metres. A car can keep it up for hours.
  • Sound depends on the air: in warmer air it is a little faster, in colder air a little slower. 343 m/s is the usual figure for 20 °C.
  • Light is the fastest thing there is. In one second it could go round the Earth more than 7 times.

A ladder of speeds

The speeds in the table are so different that no ordinary ruler can show them. One trick is to give each power of ten an equal step: 1 m/s, 10, 100, 1000, and so on, each step ten times the one below.

1 billion1 million100011 thousandth1 millionth1 billionth m/s light: about 300 000 000 Earth round the Sun: about 30 000 sound in air: 343 jet airliner: about 250 cheetah: about 28 Usain Bolt: about 10 walking: 1.4 Curiosity rover: 0.04 hair growing: about 0.000 000 004

On this ladder, walking and the rover are about one and a half rungs apart (a walker is 35 times faster), and light is more than six rungs above sound (almost a million times faster). Writing such huge and tiny numbers neatly is the job of Scientific notation.

Comparing: how many times faster?

To compare two speeds in the same units, you can subtract (how much faster) or divide (how many times as fast). Dividing is usually more telling for very different speeds.

  • Walker and rover: \(1.4 \div 0.04 = 35\). A walker is 35 times as fast.
  • Cheetah and walker: \(28 \div 1.4 = 20\). A sprinting cheetah is about 20 times as fast as you walk.
  • Light and sound: \(300\,000\,000 \div 343\), about 870,000. Light is nearly a million times faster.

Worked examples

1. Walking in km/h. A person walks at 1.4 m/s. In km/h: \(1.4 \times 3.6 = 5.04\), about 5 km/h. That is why people say "an hour's walk is about 5 km".

2. Cheetah against Bolt. In m/s, a cheetah sprints at roughly 28 and Usain Bolt averaged about 10.4. In km/h: \(28 \times 3.6\) is about 100, and \(10.4 \times 3.6\) is about 37.5. The cheetah is about \(28 \div 10.4\), roughly 2.7 times as fast as the fastest human ever timed.

3. How far away is the storm? You see lightning, then hear the thunder 6 seconds later. Light reaches you almost instantly (in a few millionths of a second), but sound takes time. In 6 seconds, sound goes \(343 \times 6 = 2058\) metres. The storm is about 2 km away. A handy rule: every 3 seconds is about 1 km, because \(343 \times 3 = 1029\) metres.

4. The rover's long drive. At its top speed of 4 cm/s, how long would Curiosity take to drive 1 km without stopping?

1 km is 100,000 cm. \(100\,000 \div 4 = 25\,000\) seconds. In hours: \(25\,000 \div 3600\), about 6.9 hours. In practice it drives much more slowly than its top speed, stopping often to check its path and take pictures, and it has travelled only a few tens of kilometres in many years on Mars.

5. A message to Mars. When Mars is 225 million km away, how long does a radio message (travelling at the speed of light, about 300,000 km/s) take to get there?

\(225\,000\,000 \div 300\,000 = 750\) seconds, which is 12.5 minutes. Depending on where the two planets are in their orbits, the delay is anything from about 3 minutes to about 22 minutes. That is why nobody can drive a Mars rover live with a joystick.

Try it

  1. Change 90 km/h into m/s.
  2. Change 15 m/s into km/h.
  3. Which is faster: 72 km/h or 18 m/s?
  4. You hear thunder 9 seconds after the lightning. About how far away is the storm?
  5. Walking at 1.4 m/s, how long does it take to walk 3 km? Give your answer in minutes.
  6. Which is faster, a jet at 900 km/h or sound at 343 m/s? By how many km/h?
  7. Sunlight travels about 150 million km to reach Earth. How long does it take, at 300,000 km/s?
  8. Challenge: hair grows about 1 cm a month. How many millimetres per day is that (use 30 days in a month)? About how many times faster is a walker (1.4 m/s) than growing hair (about 0.000 000 004 m/s)?
Answers
  1. \(90 \div 3.6 = 25\) m/s.
  2. \(15 \times 3.6 = 54\) km/h.
  3. \(72 \div 3.6 = 20\) m/s, which is more than 18 m/s. So 72 km/h is faster.
  4. \(343 \times 9 = 3087\) metres: about 3 km (9 seconds is 3 lots of 3 seconds).
  5. 3 km is 3000 m. \(3000 \div 1.4\) is about 2143 seconds. \(2143 \div 60\) is about 35.7 minutes, roughly 36 minutes. (Check with 5 km/h: 3 km takes \(\frac{3}{5}\) of an hour, which is 36 minutes. ✓)
  6. Sound: \(343 \times 3.6\), about 1235 km/h. Sound is faster, by about \(1235 - 900 = 335\) km/h. (A jet that flew faster than sound would be "supersonic".)
  7. \(150\,000\,000 \div 300\,000 = 500\) seconds, which is 8 minutes 20 seconds. The sunlight you see left the Sun about 8 minutes ago.
  8. 1 cm is 10 mm, so \(10 \div 30\), about 0.33 mm per day. Walker against hair: \(1.4 \div 0.000\,000\,004\), which is \(1.4 \times 250\,000\,000 = 350\,000\,000\). About 350 million times faster. (With a more exact hair speed, about 0.000 000 003 9 m/s, it is about 360 million.)

Common mistake

Multiplying when you should divide. To change km/h into m/s, divide by 3.6; to change m/s into km/h, multiply. Check your answer: the km/h number must be bigger than the m/s number for the same speed.

Comparing numbers without units. "The rover does 4 and a walker does 1.4, so the rover is faster" is wrong: the rover's 4 is centimetres per second, only 0.04 m/s, and the walker is 35 times faster. Convert to the same units first, then compare.

Where it leads