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 $v_{\max}=\sqrt{2 a s}$
$v_{avg}=\frac{6 s}{ T}T=t_{1}+t_{2}+t_{3}$
$t_{1}=\sqrt{\frac{2 s}{a}},t_{2}=\frac{2 s}{v_{\max}}=\sqrt{\frac{2 s}{a}}$
Retardation $\Rightarrow \beta $
$\beta =\frac{v_{\max}^{2}}{6 s}$
$\Rightarrow \beta =\frac{2 a \times s}{6 \times s}=\frac{a}{3}$
$t_{3}=\frac{v_{\max}}{\beta }=3\sqrt{\frac{2 a \times s}{a}}=3\sqrt{\frac{2 s}{a}}$
$T=5\sqrt{\frac{2 s}{a}}$
$v_{avg}=\frac{6}{5}\frac{5}{\sqrt{\frac{2 s}{a}}}$
$v_{avg}=\frac{6}{5}\sqrt{\frac{a s}{2}}$
$\frac{v_{avg}}{v_{\max}}=\frac{6}{5}\sqrt{\frac{a s}{2}}\times \frac{1}{\sqrt{2 a s}}=\frac{3}{5}$