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Q.
A particle starts from rest with constant acceleration. The ratio of space-average velocity to the time average velocity is :
Motion in a Straight Line
Solution:
$v=a t, \frac{d s}{d t}=a t \Rightarrow d s=a t d t$
$\left(v_{\text {avg }}\right)_{\text {space }}=\frac{\int v d s}{\int d s}=\frac{\int \text { at at } d t}{\int a t\, d t}$
$=\frac{a^{2} \frac{t^{3}}{3}}{\frac{a t^{2}}{2}}=\frac{2 a t}{3}$
$\left(v_{\text {aug }}\right)_{\text {time }}=\frac{\int v d t}{\int d t}=\frac{\int a t d t}{\int d t}=\frac{a^{t^{2}}{2}}{t}=\frac{\text { at }}{2}$
$\frac{\left( v _{ avg }\right)_{\text {space }}}{\left( v _{\text {avg }}\right)_{\text {time }}}=\frac{\frac{2 at }{3}}{\frac{ at }{2}}=\frac{4}{3}$