## Task 20/64

A racing car drives a test track $s$ three times in a flying start, with the speeds $v_{1}, v_{2}$ and $v_{3}$ being measured $\left(v_{1} \neq v_{2} \neq v_{3}\right)$. What is the average speed $v_{m}$ from the three test runs?
The average speed $v_{m}$ is obtained by dividing the sum of all distances traveled by the sum of the required times:

$$
v_{m}=\frac{\sum_{i=1}^{n} s_{i}}{\sum_{i=1}^{n} t_{i}}
$$

Now, $s-1=s_{2}=s_{3}=s$, so $\sum_{i=1}^{3} s_{i}=3 s$. From $v=\frac{s}{t}$, it follows that $t_{i}=\frac{s_{i}}{v_{i}}=\frac{s}{v_{i}}$. Therefore,

$$
\sum_{i=1}^{3} t_{i}=\sum_{i=1}^{3} \frac{s}{v_{i}}=s\left(\frac{1}{v_{1}}+\frac{1}{v_{2}}+\frac{1}{v_{3}}\right)
$$

For $v_{m}$, it follows that

$$
v_{m}=\frac{3 s}{s\left(\frac{1}{v_{1}}+\frac{1}{v_{2}}+\frac{1}{v_{3}}\right)}=3 \frac{v_{1} v_{2} v_{3}}{v_{1} v_{2}+v_{2} v_{3}+v_{1} v_{3}}
$$

If we take the reciprocal value $\frac{1}{v_{m}}$, we get

$$
\frac{1}{v_{m}}=\frac{1}{3}\left(\frac{1}{v_{1}}+\frac{1}{v_{2}}+\frac{1}{v_{3}}\right)
$$

We can see that the average speed is not equal to the arithmetic mean of the individual speeds, but rather it is equal to the so-called harmonic mean of them. The reciprocal value of the average speed is equal to the arithmetic mean of the reciprocal values of the individual speeds.