Why Two Equal Distances Do Not Give Two Equal Votes on Speed

By Yossi Gerson

Imagine a walk with two equal sections. You cover the first two kilometers at four kilometers per hour and the next two at two kilometers per hour. What is your average speed for the whole walk?

Three kilometers per hour is a tempting answer: add four and two, then divide by two. But the two sections take different amounts of time. The slower section occupies more of the walk, so it has more influence on the average.

The examples below use invented journeys and constant speeds within each section. They are arithmetic models, not travel estimates.

Start with distance and time

Average speed describes the distance covered per unit of elapsed time over a chosen interval. OpenStax's Speed and Velocity gives the definition and distinguishes it from instantaneous speed and average velocity.

For a whole journey, use:

Average speed = total distance divided by total elapsed time.

The first section is two kilometers at four kilometers per hour. Its time is two divided by four, or half an hour. The second section is two kilometers at two kilometers per hour. It takes one hour.

Section	Distance	Speed	Time
First	2 km	4 km/h	0.5 h, or 30 minutes
Second	2 km	2 km/h	1 h, or 60 minutes
Whole walk	4 km	To calculate	1.5 h, or 90 minutes

Divide four kilometers by one and a half hours. The average is 8/3 kilometers per hour, approximately 2.67 km/h.

You can test the tempting answer of 3 km/h. At that average, four kilometers would take 4/3 hours, or 80 minutes. Our two sections actually require 90 minutes. An answer that changes the journey's total time cannot describe this journey.

The missing weight is time

The first speed applies for 30 minutes; the second applies for 60. In terms of time, the slower section gets twice the weight.

Multiplying each section's speed by its duration recovers its distance:

4 km/h × 0.5 h = 2 km.

2 km/h × 1 h = 2 km.

Add those distances and divide by the combined duration. This gives the same 2.67 km/h. The multiplication is useful because it shows what a weighted average is doing. Each speed contributes according to how long it lasts.

Equal distances do not supply equal time weights. A slower speed needs more time to cover its share of the distance.

When the simple average does work

Change the example so that each speed lasts half an hour.

At four kilometers per hour, the first half-hour covers two kilometers. At two kilometers per hour, the second half-hour covers one kilometer. Now the total is three kilometers in one hour, so the average speed is three kilometers per hour.

Here, adding the two speeds and dividing by two works because their durations are equal. The difference between the examples is what was held equal: distance in the first, time in the second.

For several sections, adding the speeds and dividing by the number of sections works when every section has the same duration. Otherwise, recover each distance and use the actual total time.

Decide what happens to a stop

Suppose the original four-kilometer walk includes a 15-minute stop between its two sections. If you measure from departure to arrival, the stop belongs in elapsed time.

The distance remains four kilometers. The duration becomes 105 minutes, or 1.75 hours. The overall average is therefore 4 divided by 1.75, approximately 2.29 km/h.

If you deliberately exclude the stop, you are describing the average during moving time: approximately 2.67 km/h. Both calculations can be useful, but their labels must make the time interval clear. An app's reported average is hard to compare with your own unless you know how it treats pauses.

Speed and velocity answer different questions

Average speed uses the full distance traveled. Average velocity divides displacement, the change from starting position to ending position, by elapsed time. It is a vector, so direction matters.

A walk that returns to its starting point has zero displacement. Its average velocity over the complete trip is zero, although its average speed is positive if it covered a positive distance in a positive time. Returning home does not erase the distance walked.

To see distance and displacement on a graph, read Read motion graphs without guessing. Its worked journey includes a pause and a reversal, so average speed and average velocity answer visibly different questions.

Try the method on one more journey

A hypothetical cyclist covers three kilometers at six kilometers per hour, then three kilometers at three kilometers per hour, with no stops.

The two times are half an hour and one hour. Six kilometers divided by 1.5 hours gives 4 km/h, rather than the simple average of 4.5 km/h.

For your own calculation, write down three things before dividing: the distance of each section, the duration of each section, and whether the interval includes stops. Convert the durations to the same unit. Add distances, add times, then divide once.

That short record makes the result explainable. More importantly, it tells you which journey the number actually describes.