Question

A body covers first one-third of its journey with speed $u$, next one- third with speed $v$ and the last one-third with speed $w$. Calculate the average speed of the body over the entire journey.

• A. $\frac{uvw}{uv+vw+uw}$
• B. $\frac{3uvw}{uv+vw+uw}$
• C. $\frac{2uvw}{uv+vw+uw}$
• D. $\frac{4uvw}{uv+vw+uw}$

Answer 1

The correct option is B $\frac{3uvw}{uv+vw+uw}$

Let the distance travelled in the whole journey be $D$
Part 1: (One third distance with speed $u$)
$\text{Distance}=\frac{D}{3}$; $\text{Speed}=u$
$\text{Time}=\frac{\text{Distance}}{\text{Speed}}=\frac{D}{3u}$
Part 2: (One third with speed $v$)
$\text{Distance}=\frac{D}{3}$; $\text{Speed}=v$
$\text{Time}=\frac{\text{Distance}}{\text{Speed}}=\frac{D}{3v}$
Part 3: (One third with speed $w$)
$\text{Distance}=\frac{D}{3}$; $\text{Speed}=w$
$\text{Time}=\frac{\text{Distance}}{\text{Speed}}=\frac{D}{3w}$.
Average speed over the entire journey $=\frac{\text{Total distance}}{\text{Total time}}$
$=\frac{D}{\frac{D}{3u}+\frac{D}{3v}+\frac{D}{3w}}=\frac{3uvw}{uv+vw+uw}$