# A body covers a certain distance x in equal thirds - the first third of distance with speed v$_1$, the second third with a speed of v$_2$ and the last third with a speed of v$_3$. What is the average speed of the body over time?

Answer: $\large\frac{3v_1v_2v_3}{v_2v_3+v_1v_3 + v_1v_2}$
In general terms, the time taken to cover a third of the distance $x$ with speed $v$ is $t = \large\frac{\text{x}/3}{\text{v}}$
$\Rightarrow$ the overall speed $\overline{v} = \Large\frac{x}{ \frac{x}{3v_1} + \frac{x}{3v_2} + \frac{x}{3v_3}}$
$\Rightarrow \overline{v} = \large\frac{3v_1v_2v_3}{v_2v_3+v_1v_3 + v_1v_2}$

An easier way to solve this is by using a simpler formula which is used for bodies travelling distances in equal intervals of time: Avg.speed= 1÷(1/n(1/v1+1/v2+...1/vn) n=no.of intervals. 1÷(1/3(1/v1+1/v2+1/v3)) 3v1v2v3/v1v2+v2v3+v3v1