Dec 7, 2017

Space average velocity $= \frac{\int v \mathrm{dx}}{\int \mathrm{dx}}$ ......(1) (for one dimensional motion)
Time average velocity $= \frac{\int v \mathrm{dt}}{\int \mathrm{dt}}$ .......(2)

Applicable kinematic expressions are

$v = u + a t$
$s = u t + \frac{1}{2} a {t}^{2}$

Given that particle starts from rest with constant acceleration. WE have

$v = a t$ .....(3)
$x = \frac{1}{2} a {t}^{2}$ ......(4)

From (3) and (4) $v$ in terms of $x$ is

$v = a \times \sqrt{2 \frac{x}{a}}$
$\implies v = \sqrt{2 a x}$ ......(5)

From (1)
Space average velocity $= \frac{{\int}_{0}^{x} \sqrt{2 a x} \mathrm{dx}}{{\int}_{0}^{x} \mathrm{dx}} = \frac{\sqrt{2 a} {x}^{\frac{3}{2}} / \left(\frac{3}{2}\right)}{x} = \frac{2}{3} \sqrt{2 a x}$

Space average velocity$= \frac{2}{3} v$ .......(6)

From (3)
Time average velocity $= \frac{\int v \mathrm{dt}}{\int \mathrm{dt}} = \frac{{\int}_{0}^{t} a t \mathrm{dt}}{{\int}_{0}^{t} \mathrm{dt}} = \frac{\frac{1}{2} a {t}^{2}}{t} = \frac{1}{2} a t$

Time average velocity$= \frac{1}{2} v$ .......(7)

Required Ratio $= \text{space average velocity"/ "time average velocity} = \frac{\frac{2}{3} v}{\frac{1}{2} v} = \frac{4}{3}$