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Speed as a rate

Ratio, rates and money · level 3, lesson 5 of 6

A question to start

In 2009 the sprinter Usain Bolt ran 100 metres in 9.58 seconds, a world record that still stood many years later. A city bus drives along a street at 40 kilometres per hour.

Who is faster over that race: the fastest sprinter in history, or an ordinary bus?

It feels like the sprinter should win. But the two speeds are written in different units, metres in seconds against kilometres in hours, so you cannot compare them by looking. In this lesson you will learn to turn one into the other, and the answer may surprise you.

The idea

Speed is a rate

Speed tells you how far something goes in each unit of time. It is a rate, distance per time:

\[\text{speed} = \frac{\text{distance}}{\text{time}}.\]

If a car goes 240 km in 3 hours, its speed is \(240 \div 3 = 80\) km per hour (written 80 km/h). If a snail crawls 6 metres in 2 minutes, its speed is 3 metres per minute.

The unit of a speed is always "a distance unit per a time unit": km/h, m/s (metres per second), m/min. The unit tells you which numbers were divided.

Steady speed and average speed

Real things rarely move at a perfectly steady speed. A car slows at lights and speeds up on the highway. When we say "240 km in 3 hours is 80 km/h", we mean the average speed: the steady speed that would cover the same distance in the same time. In this lesson, "speed" means average speed unless we say otherwise. (The speed at one single instant, the number on a speedometer, belongs to Change.)

Three facts in one

Speed, distance and time are tied together. Know any two and you can find the third.

\[\text{distance} = \text{speed} \times \text{time}\]
\[\text{time} = \frac{\text{distance}}{\text{speed}}\]

Why does distance equal speed times time? A speed of 80 km/h means each hour adds 80 km. After 3 hours you have three lots of 80 km: \(3 \times 80 = 240\) km. That is just the "for every" of level 1: 80 km for every hour.

Why is time distance divided by speed? To drive 240 km at 80 km/h, ask "how many lots of 80 km fit into 240 km?" That is a division: \(240 \div 80 = 3\) hours.

Many people remember the three facts with a triangle. Cover the one you want; what is left shows how to get it.

distance speed time ×

Cover "distance": you see speed × time. Cover "time": you see distance over speed. The triangle is only a memory aid; the reasons above are what make it true.

Units must match

The units must fit together. If the speed is in km/h, the time must be in hours and the distance comes out in kilometres. Minutes must be changed to parts of an hour first:

Minutes 15 20 30 45 6
Hours 0.25 \(\tfrac13\) 0.5 0.75 0.1

So 2 hours 15 minutes is 2.25 hours, not 2.15 hours. (There are 60 minutes in an hour, not 100.)

Changing km/h into m/s

To compare the sprinter and the bus we need one unit. Let us find what 1 km/h is in metres per second.

  • 1 km is 1000 m.
  • 1 hour is \(60 \times 60 = 3600\) seconds.
  • So 1 km/h means 1000 m in 3600 s, which is \(1000 \div 3600 = \tfrac{1}{3.6}\) m/s.

That gives a handy rule:

km/h to m/s: divide by 3.6. m/s to km/h: multiply by 3.6.

For example, 36 km/h is \(36 \div 3.6 = 10\) m/s, and 10 m/s is \(10 \times 3.6 = 36\) km/h.

Now the race. Bolt's average speed was \(100 \div 9.58 \approx 10.4\) m/s. In km/h: \(10.4 \times 3.6 \approx 37.6\) km/h. The bus at 40 km/h is a little faster than the fastest human ever, averaged over that race. (Bolt was faster than the bus near the end of his run, when he was at full speed, but he started from standing still.)

Average speed for a whole trip

For a trip with several parts, the average speed is

\[\text{average speed} = \frac{\text{total distance}}{\text{total time}}.\]

It is not the average of the speeds. You will see why in worked example 6.

Speed on Mars

Speeds on Mars show how big the numbers can get. NASA's Curiosity rover can drive at about 4 cm per second on flat, hard ground, slower than a walking tortoise. Meanwhile, a radio message from Earth to Mars travels at the speed of light, about 300 000 km every second. Even so, Mars is so far away (from about 55 million km to about 400 million km, depending on where the two planets are in their orbits) that a message takes several minutes to arrive.

Worked examples

1. Finding distance. A cyclist rides at 15 km/h for 2.5 hours. How far?

\(\text{distance} = 15 \times 2.5 = 37.5\) km.

2. Finding time. A bus travels 210 km at an average of 70 km/h. How long does it take?

\(\text{time} = 210 \div 70 = 3\) hours.

3. Finding speed with minutes. Lee walks 9 km in 2 hours 15 minutes. What is her average speed in km/h?

  • 15 minutes is \(\tfrac{15}{60} = 0.25\) hours, so the time is 2.25 hours.
  • \(\text{speed} = 9 \div 2.25 = 4\) km/h.

4. Changing units. A car drives at 72 km/h. A cheetah runs at 25 m/s. Which is faster?

  • Car in m/s: \(72 \div 3.6 = 20\) m/s.
  • The cheetah at 25 m/s is faster. (Or: cheetah in km/h is \(25 \times 3.6 = 90\) km/h.)

5. A message to Mars. How long does a radio message take to reach Mars when Mars is 55 million km away? And 400 million km?

  • \(\text{time} = 55\,000\,000 \div 300\,000 \approx 183\) seconds, about 3 minutes.
  • \(\text{time} = 400\,000\,000 \div 300\,000 \approx 1333\) seconds, about 22 minutes.

This is why a rover cannot be steered with a joystick from Earth: the picture of a rock in its way would arrive minutes late, and the command "stop!" would take minutes more to get back.

6. The average-speed trap. You cycle 12 km to a lake at 24 km/h, and come back the same 12 km at 12 km/h because it is uphill. What is your average speed?

  • Time there: \(12 \div 24 = 0.5\) hours. Time back: \(12 \div 12 = 1\) hour.
  • Total: 24 km in 1.5 hours.
  • Average speed: \(24 \div 1.5 = 16\) km/h, not 18 km/h (the average of 24 and 12).

You spend twice as long going slowly as going fast, so the slow speed counts more.

Try it

  1. A car drives for 3 hours at 60 km/h. How far does it go?
  2. A boat sails 45 km at 15 km/h. How long does it take?
  3. A runner covers 400 m in 80 seconds. What is her speed in m/s? In km/h?
  4. Change 90 km/h into m/s.
  5. A scooter rides for 30 minutes at 50 km/h. How far does it go?
  6. A train leaves at 9:00 and arrives 270 km away at 12:00. What was its average speed? What time would it arrive if it averaged 90 km/h instead, still leaving at 9:00?
  7. At 4 cm per second, how long would Curiosity take to drive 100 m? Give your answer in minutes.
  8. You walk up a hill at 4 km/h and straight back down the same path at 12 km/h. What is your average speed for the whole walk? (Hint: pick a length for the path, such as 12 km.)
Answers
  1. \(60 \times 3 = 180\) km.
  2. \(45 \div 15 = 3\) hours.
  3. \(400 \div 80 = 5\) m/s. In km/h: \(5 \times 3.6 = 18\) km/h.
  4. \(90 \div 3.6 = 25\) m/s.
  5. 30 minutes is 0.5 hours. \(50 \times 0.5 = 25\) km.
  6. From 9:00 to 12:00 is 3 hours, so \(270 \div 3 = 90\) km/h. It already averaged 90 km/h, so it would arrive at 12:00, the same time. (A trick question: always check what you are given!)
  7. 100 m is 10 000 cm. \(10\,000 \div 4 = 2500\) seconds. \(2500 \div 60 \approx 41.7\), so about 42 minutes.
  8. Take the path as 12 km. Up: \(12 \div 4 = 3\) hours. Down: \(12 \div 12 = 1\) hour. Total: 24 km in 4 hours, so \(24 \div 4 = 6\) km/h. (Any length gives 6 km/h; try 24 km to check.) Not 8 km/h.

Common mistake

Treating minutes as decimals. 1 hour 30 minutes is 1.5 hours, not 1.3 hours, because 30 minutes is half of 60. Change minutes into hours by dividing by 60.

Averaging the speeds. For a trip in parts, the average speed is total distance over total time. Adding two speeds and halving works only if you spend the same time at each speed, not the same distance.

Where it leads