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Average speed for a trip

Change and motion · level 3, lesson 6 of 6

A question to start

Grandma lives 60 km away. On the way there, the traffic is terrible and you only manage 30 km/h (kilometres per hour). On the way home you want to make up for it, so that your average speed for the whole round trip is 60 km/h.

How fast must you drive home?

Most people say 90 km/h, because 30 and 90 average to 60. Try it at the end of this lesson. The real answer is very surprising: no speed at all is fast enough, not even the speed of a rocket.

The idea

What "average speed" means

Almost nothing moves at exactly the same speed all the time. A car speeds up, slows down, stops at lights. So when we say "the trip was at 50 km/h", we usually mean the average speed:

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

It answers this question: if you had gone at one steady speed the whole time, what speed would cover the same distance in the same time?

For example, a bus goes 150 km in 3 hours. It sped up and slowed down along the way, but a steady 50 km/h would also cover 150 km in 3 hours, since \(150 \div 3 = 50\). So its average speed is 50 km/h.

This is just the "per" idea of lesson 1: total distance shared fairly over the total time.

There and back: why it is not 45

You drive 60 km to a lake at 30 km/h, and 60 km back at 60 km/h. What is your average speed?

The tempting answer is "halfway between 30 and 60, so 45". Let us actually work it out.

  • Going there: 60 km at 30 km/h takes \(60 \div 30 = 2\) hours.
  • Coming back: 60 km at 60 km/h takes \(60 \div 60 = 1\) hour.
  • Total: 120 km in 3 hours.
  • Average speed: \(120 \div 3 =\) 40 km/h.

Not 45. Where did 45 go wrong? Look at how the time was spent:

hour 1hour 2hour 3 30 km30 km60 km going: slowcoming back: fast 120 km in 3 hours: 40 km in each hour, on average

You spent two hours at the slow speed but only one hour at the fast speed. The slow speed went on for longer, so it counts for more. Hour by hour, your speeds were 30, 30 and 60. The fair average of those three hours is

\[\frac{30 + 30 + 60}{3} = \frac{120}{3} = 40.\]

That is the real reason. "Averaging the two speeds" pretends you spent the same time at each speed, but you did not. Going slowly makes a trip take longer, so slow parts always fill up more of the time.

When averaging the speeds does work

If you spend equal times at each speed, then averaging the speeds is fine. One hour at 30 km/h and one hour at 60 km/h covers \(30 + 60 = 90\) km in 2 hours: \(90 \div 2 = 45\) km/h, which is halfway between 30 and 60.

So:

  • Same time at each speed: the average speed is halfway between (the average of the speeds).
  • Same distance at each speed: the average speed is less than halfway, closer to the slow speed, because more time is spent going slowly.

A picture on a graph

Here is the lake trip as a graph of total distance travelled against time.

0123 time (hours) 0306090120 km 30 km/h60 km/haverage: 40 km/h

The orange line is the real trip: gentle for two hours, then steep for one. The dashed green line goes straight from the start to the end. It is the trip at one steady speed that ends at the same place at the same time. Its steepness is the average speed: 120 km in 3 hours, 40 km/h.

Notice the dashed line is closer in steepness to the long, gentle part. The long part has more say.

Stops count too

The clock does not stop when you stop. If you drive for 2 hours and then have lunch for 1 hour, the lunch hour is part of the total time. A stop is a part of the trip at speed 0, and it pulls the average down.

Back to Grandma

Now the puzzle. The round trip is \(60 + 60 = 120\) km. To average 60 km/h, the whole trip must take \(120 \div 60 = 2\) hours.

But getting there at 30 km/h already took \(60 \div 30 = 2\) hours. All your time is used up before you even turn around. To average 60 km/h you would have to drive the 60 km home in no time at all. No car, plane or rocket can do that. The slow first half has eaten the whole time budget.

Worked examples

1. A bus trip. A bus travels 150 km in 3 hours. Average speed: \(150 \div 3 = 50\) km/h.

2. A trip with a stop. A family drives 120 km in 1.5 hours, stops for lunch for 30 minutes, then drives 60 km in 1 hour.

  • Total distance: \(120 + 60 = 180\) km.
  • Total time: \(1.5 + 0.5 + 1 = 3\) hours (the lunch counts).
  • Average speed: \(180 \div 3 = 60\) km/h.

While driving, they did 80 km/h and then 60 km/h. The lunch stop pulled the average down to 60.

3. Two laps of a track. A runner does two laps of a 400 m track. The first lap takes 80 seconds and the second, tired, takes 100 seconds.

  • Total: 800 m in 180 seconds.
  • Average speed: \(800 \div 180\), which is about 4.4 metres per second.

Each lap on its own: \(400 \div 80 = 5\) m/s, and \(400 \div 100 = 4\) m/s. The average of 5 and 4 would be 4.5, but the true average is about 4.44, a bit lower, because the slow lap took longer.

4. Equal times. A walker goes at 6 km/h for 1 hour and then at 4 km/h for 1 hour. The times are equal, so the average speed is halfway: 5 km/h. Check: \((6 + 4) \div 2 = 5\) km/h, and \(10 \text{ km} \div 2 \text{ h} = 5\) km/h. ✓

Try it

  1. A car goes 240 km in 4 hours. What is its average speed?
  2. Pat walks at 4 km/h for one hour and then at 2 km/h for one hour. What is her average speed?
  3. Lin cycles 12 km to the beach at 12 km/h and comes back the same 12 km at 24 km/h. What is her average speed?
  4. A train goes 200 km in 2 hours, waits at a station for 1 hour, then goes 100 km in 1 hour. Find its average speed for the whole journey.
  5. A car drives part of a trip at 20 km/h and the rest at 50 km/h, with no stops. Which of these could be its average speed: 15, 30, 45, 55 km/h?
  6. Ali runs 2 km at 10 km/h and then 2 km at 15 km/h. Find his average speed.
  7. A Mars rover drives 60 metres in 30 minutes, stops for 90 minutes to take pictures, then drives another 60 metres in 30 minutes. What is its average speed in metres per minute? What was its speed while moving?
  8. Challenge: you drive 60 km to Grandma's at 40 km/h. How fast must you drive the 60 km home to average 48 km/h for the round trip?
Answers
  1. \(240 \div 4 = 60\) km/h.
  2. Equal times, so the average is halfway: 3 km/h. Check: 6 km in 2 hours, \(6 \div 2 = 3\).
  3. There: \(12 \div 12 = 1\) hour. Back: \(12 \div 24 = 0.5\) hours. Total: 24 km in 1.5 hours. \(24 \div 1.5 = 16\) km/h. (Not 18, the average of 12 and 24.)
  4. Total distance \(200 + 100 = 300\) km. Total time \(2 + 1 + 1 = 4\) hours. \(300 \div 4 = 75\) km/h.
  5. 30 and 45. With no stops, the average must lie between the slowest speed (20) and the fastest (50). 15 is too slow and 55 is too fast.
  6. First part: \(2 \div 10 = 0.2\) hours (12 minutes). Second: \(2 \div 15\) hours, which is 8 minutes. Total: 4 km in 20 minutes, which is \(\frac{1}{3}\) of an hour. Average: \(4 \div \frac{1}{3} = 4 \times 3 = 12\) km/h.
  7. Total distance 120 m. Total time \(30 + 90 + 30 = 150\) minutes. Average: \(120 \div 150 = 0.8\) metres per minute. While moving: \(60 \div 30 = 2\) metres per minute. The long stop pulls the average far down.
  8. The whole trip is 120 km, so to average 48 km/h it must take \(120 \div 48 = 2.5\) hours. Getting there took \(60 \div 40 = 1.5\) hours. That leaves 1 hour for the 60 km home: 60 km/h. (Not 56, which is what "averaging the speeds" would say.)

Common mistake

Averaging the speeds. For a trip in parts, adding the speeds and dividing by 2 only works when the times are equal. When the distances are equal, the slow part takes longer and counts for more. Always use total distance divided by total time.

Leaving out the stops. The average speed of a trip counts every minute from the start to the end, including lunch, traffic lights and waiting. If you want the speed while moving, say so, and leave out the stops on purpose.

Where it leads