Question

A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time average velocity is :

• A. $\frac{1}{2}$
• B. $\frac{3}{4}$
• C. $\frac{4}{3}$
• D. $\frac{3}{2}$

Answer 1

Answer: (C)

$v=at,\frac{ds}{dt}=at\Rightarrow ds=atdt$
${\left(v_{\text{avg }}\right)}_{\text{space }}=\frac{\int vds}{\int ds}=\frac{\int \text{ at at }dt}{\int atdt}$
$=\frac{a^2\frac{t^3}{3}}{\frac{a{t}^2}{2}}=\frac{2at}{3}$
${\left(v_{\text{aug }}\right)}_{\text{time }}=\frac{\int vdt}{\int dt}=\frac{\int atdt}{\int dt}=\frac{a^{t^2}2}{t}=\frac{\text{ at }}{2}$
$\frac{{\left(v_{avg}\right)}_{\text{space }}}{{\left(v_{\text{avg }}\right)}_{\text{time }}}=\frac{\frac{2at}{3}}{\frac{at}{2}}=\frac{4}{3}$