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Play: Fast and slow

Change and motion · step 1 of 5

A question to start

Mia and Leo have a race, but they cannot agree on the track.

  • Mia runs 100 metres and takes 20 seconds.
  • Leo runs 50 metres and takes 9 seconds.

Who is faster? Mia ran further, and Leo took less time, so it is not obvious. You cannot just look at one number. On this page you will find a fair way to compare, and then use it on things as slow as a robot on Mars and as fast as a rocket.

The idea

Fast means a lot of distance in a little time

Being fast means covering a lot of ground without taking much time. So speed always needs two things: how far you went, and how long it took. One number on its own can trick you.

To compare fairly, make one thing the same

The easiest way to compare two movers is to make either the distance or the time the same for both.

  • Same distance. Who would finish 100 metres first? Leo ran 50 metres in 9 seconds. If he kept going at the same pace, the next 50 metres would take another 9 seconds, so 100 metres would take him 18 seconds. Mia took 20. Leo is faster.
  • Same time. How far does each go in one second? This is what the word speed usually means: how far you go in one unit of time, like one second or one hour. A car going "100 kilometres an hour" would go 100 kilometres if it kept that speed for one whole hour.

Speed can change

Almost nothing moves at the same speed all the time. A car starts at zero, speeds up, slows down at a red light, stops, and starts again. A ball you throw up slows down, stops for a moment at the top, and speeds up on the way down. When we say "the car went 60 kilometres in an hour", we are really talking about its speed on average; at some moments it was faster and at some moments slower.

You can tell if something is speeding up: in each second it goes further than in the second before. If it goes less far each second, it is slowing down.

Steepness is a kind of speed too

Hills and ramps change height as you move along them. A steep ramp goes up a lot while you move only a little bit forward. A gentle ramp goes up only a little.

A: up 2 foreach 1 along B: up 1 foreach 2 along

Ramp A climbs 2 squares for every 1 square along. Ramp B climbs only 1 square for every 2 squares along. Ramp A is much steeper. Later in this topic, steepness and speed turn out to be the same idea: speed is how quickly distance changes as time goes on, and steepness is how quickly height changes as you go along.

How fast are things?

Some real speeds, to get a feel for them:

  • A person walking slowly covers about 1 metre each second.
  • NASA's Curiosity rover on Mars drives at about 4 centimetres each second at most. A slow walk is about 25 times faster. The rover is slow on purpose: it is far from any mechanic, and it must not hit a rock it has not seen.
  • To leave Earth and never fall back, a rocket must reach about 11 kilometres every second. That is fast enough to cross a big city in a few seconds. To leave Mars, which is smaller and pulls less strongly, about 5 kilometres every second is enough.

Worked example

A bike ride. Ria writes down how far she has ridden at the end of each minute.

Minute 1 2 3 4 5
Total distance (metres) 200 450 750 900 900

What happened in each minute?

  1. Minute 1: from 0 to 200 metres. She rode 200 metres.
  2. Minute 2: from 200 to 450. She rode 250 metres, more than the first minute, so she sped up.
  3. Minute 3: from 450 to 750. She rode 300 metres. Faster again.
  4. Minute 4: from 750 to 900. Only 150 metres. She slowed down, maybe going up a hill.
  5. Minute 5: from 900 to 900. No distance at all. She stopped.

The trick is to look at the changes between the numbers, not the numbers themselves. The total distance only ever goes up, but the speed went up and then down. Her fastest minute was minute 3.

Try it

  1. Ana walks 6 kilometres in 2 hours. Ben walks 9 kilometres in 3 hours. Who walks faster?
  2. A toy car goes 30 centimetres every second. How far does it go in 5 seconds?
  3. Sam rides his bike 2 kilometres in the first 10 minutes, and 1 kilometre in the next 10 minutes. Did he speed up or slow down?
  4. Ramp C goes up 3 for every 6 along. Ramp D goes up 2 for every 3 along. Which is steeper?
  5. The Curiosity rover drives 4 centimetres every second. How far can it go in one minute without stopping?
Answers
  1. The same. Ana walks 3 kilometres each hour (6 split into 2 hours). Ben also walks 3 kilometres each hour (9 split into 3 hours).
  2. 30 + 30 + 30 + 30 + 30 = 150 centimetres, which is 1.5 metres.
  3. He slowed down: he went less far in the second 10 minutes than in the first.
  4. Make the "along" the same. Ramp C goes up 3 in 6 along. Ramp D goes up 2 in 3 along, so in 6 along it goes up 4. Going up 4 beats going up 3, so ramp D is steeper.
  5. A minute is 60 seconds, so 60 lots of 4 centimetres: 240 centimetres, which is 2.4 metres. About the length of a sofa and a half.

Common mistake

Thinking the one who went further was faster. Going further only means faster if the time was the same. A snail that crawls all day can go further than a cheetah that runs for two seconds. Always ask both questions: how far? and how long?

Where it leads