Question

A body moves in a straight line with speed $v_{1}$ and $v_{2}$ for distance which are in ratio $1: 2$. Find average speed.

• A. $3 v_{1}+v_{2}$
• B. $v_{2}+2 v_{1}$
• C. $\frac{3 v_{1} v_{2}}{v_{2}+2 v_{1}}$
• D. $\frac{3 v_{1} v_{2}}{v_{2}-2 v_{1}}$

Answer 1

Answer: (C)

Given that, the distances travelled by the body are in ratio $1: 2$.
Let us suppose that $s$ is the distance, the body travels with speed $v_{1}$.
Then, $2 s$ will be the distance travelled by the body with speed $v_{2} .$
This scenario is shown in the line-diagram below

For $A$ to $C$,
Average speed $=\frac{\text { Total distance travelled }}{\text { Total time taken }}$
$v_{\text {avg }}=\frac{s_{1}+s_{2}}{t_{1}+t_{2}}=\frac{s+2 s}{\frac{s}{v_{1}}+\frac{2 s}{v_{2}}}$
$=\frac{3 s}{s\left[\frac{1}{v_{1}}+\frac{2}{v_{2}}\right]}=\frac{3}{\left[\frac{v_{2}+2 v_{1}}{v_{1} v_{2}}\right]}=\frac{3 v_{1} v_{2}}{v_{2}+2 v_{1}}$