Formula

Average speed over two equal distances

Arithmetic • Time, Speed and Distance • Average Speed

Formula

average speed = \frac{2 uv}{u + v}

When the same distance is covered at speed

u

one way and

v

the other, the average speed for the whole journey is

\frac{2 uv}{u + v}

; the arithmetic mean

\frac{u + v}{2}

is the average only when the two speeds are held for equal times, not equal distances.

How to use it

1Use it whenever the outward and return legs are the same distance at different speeds.
2The underlying computation is total distance over total time, $\frac{2 d}{\frac{d}{u} + \frac{d}{v}}$ ; the formula only packages it.
3For three equal legs the pattern continues as $\frac{3}{\frac{1}{u} + \frac{1}{v} + \frac{1}{w}}$ .

A cyclist covers

30

km at

15

km/hr and returns over the same road at

10

km/hr. What is the average speed for the whole ride?

Step-by-step

1
$\frac{2 \times 15 \times 10}{15 + 10} = \frac{300}{25} = 12$ km/hr
Why: The two legs are equal in distance, so the harmonic mean applies.
2
Check: $\frac{30}{15} + \frac{30}{10} = 5$ hours for $60$ km, which is $12$ km/hr

Answer

12

km/hr

Watch out

The harmonic mean is always below the arithmetic mean, so an answer equal to

\frac{u + v}{2}

is a sign the equal-distance condition was overlooked.

Builds on

Also known as

average speed to and fro, harmonic mean of two speeds, 2uv/(u+v), round trip average speed, same distance at two speeds