Uphill and downhill

The cyclist moves uphill at a constant speed of v1 = 10 km/h. When he reaches the top of the hill, he turns and passes the same track downhill at a speed of v2 = 40 km/h. What is the average speed of a cyclist?

Correct answer:

v =  16 km/h

Step-by-step explanation:

$$\begin{gathered} v_1=10\text{km/h} \\ {v}_2=40\text{km/h} \\ s=v_1{t}_1=v_2{t}_2 \\ {t}_2=\frac{v_1}{v_2}\cdot t_1 \\ v=\frac{s+s}{t_1+t_2}=\frac{v_1\cdot t_1+v_2\cdot t_2}{t_1+t_2} \\ v=\frac{v_1\cdot t_1+v_2\cdot \frac{v_1}{v_2}\cdot t_1}{t_1+\frac{v_1}{v_2}\cdot t_1} \\ v=\frac{v_1+v_2\cdot \frac{v_1}{v_2}}{1+\frac{v_1}{v_2}} \\ v=\frac{v_1+v_1}{1+\frac{v_1}{v_2}} \\ v=\frac{2\cdot v_1\cdot v_2}{v_2+v_1}=\frac{2\cdot 10\cdot 40}{40+10}=16\text{km/h} \end{gathered}$$

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