Question

A particle starts from rest and travels a distance $s$ with uniform acceleration, then it travels a distance $2s$ with uniform speed finally it travels a distance $3s$ with uniform retardation and comes to rest. If the complete motion of the particle is a straight line then the ratio of its average velocity to maximum velocity is

• A. $6/7$
• B. $4/5$
• C. $3/5$
• D. $2/5$

Answer 1

Answer: (C)

Maximum velocity $v_{\max }=\sqrt{2as}$
$v_{avg}=\frac{6s}{T}T=t_1+t_2+t_3$
$t_1=\sqrt{\frac{2s}{a}},t_2=\frac{2s}{v_{\max }}=\sqrt{\frac{2s}{a}}$
Retardation $\Rightarrow \beta$
$\beta =\frac{v_{\max }^2}{6s}$
$\Rightarrow \beta =\frac{2a\times s}{6\times s}=\frac{a}{3}$
$t_3=\frac{v_{\max }}{\beta }=3\sqrt{\frac{2a\times s}{a}}=3\sqrt{\frac{2s}{a}}$
$T=5\sqrt{\frac{2s}{a}}$
$v_{avg}=\frac{6}{5}\frac{5}{\sqrt{\frac{2s}{a}}}$
$v_{avg}=\frac{6}{5}\sqrt{\frac{as}{2}}$
$\frac{v_{avg}}{v_{\max }}=\frac{6}{5}\sqrt{\frac{as}{2}}\times \frac{1}{\sqrt{2as}}=\frac{3}{5}$