Question

A body starts from rest and travels a distance $s$ with uniform acceleration, then covers a distance $2s$ uniformly and finally comes to rest after moving further $5s$ distance under uniform retardation. The ratio of the average velocity to maximum velocity is

• A. $\frac{2}{5}$
• B. $\frac{3}{5}$
• C. $\frac{4}{7}$
• D. $\frac{5}{7}$

Answer 1

The correct option is C $\frac{4}{7}$

Let $t_1$, $t_2$ and $t_3$ be the time taken to cover the distances $s$, $2s$ and $5s$ respectively.
Graphically, the area of $v-t$ curve represent displacement.
$s=\frac{1}{2}{v}_{\text{max}}{t}_1$ or $t_1=\frac{2s}{v_{\text{max}}}$
$2s=v_{\text{max}}{t}_2$ or $t_2=\frac{2s}{v_{\text{max}}}$
$5s=\frac{1}{2}{v}_{\text{max}}{t}_3$ or $t_3=\frac{10s}{v_{\text{max}}}$

Average velocity, $v_{av}=\frac{\text{Total displacement}}{\text{Total time}}$
$=\frac{s+2s+5s}{\frac{2s}{v_{\text{max}}}+\frac{2s}{v_{\text{max}}}+\frac{10s}{v_{\text{max}}}}$
$\frac{v_{av}}{v_{\text{max}}}=\frac{8s}{14s}=\frac{4}{7}$