Question

A point traversed half the distance with a velocity $v_0$. The remaining part of the distance was covered with velocity $v_1$ for half the time, and with velocity $v_2$ for the other half of the time. If the mean velocity of the point averaged over the whole time of motion is $⟨v⟩=\frac{x{v}_0(v_1+v_2)}{2v_0+v_1+v_2}$. Find $x$.

Answer 1

Let $s$ be the total distance traversed by the point and $t_1$ the time taken to cover half the distance. Further let $2t$ be the time to cover the rest half of the distance.
Therefore $\frac{s}{2}=v_0{t}_1$ or $t_1=\frac{s}{2v_0}$ (1)
and $\frac{s}{2}=(v_1+v_2)t$ or $2t=\frac{s}{v_1+v_2}$ (2)
Hence the sought average velocity
$⟨v⟩=\frac{s}{t_1+2t}=\frac{s}{\left[\frac{s}{2v_0}\right]+\left[\frac{s}{v_1+v_2}\right]}=\frac{2v_0(v_1+v_2)}{v_1+v_2+2v_0}$