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

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

(a) 1/2

(b) 3/4

(c) 4/3

(d) 3/2

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Ans: (c)
Sol: Space average velocity , $\displaystyle = \frac{\int v dx}{\int dx}$

Time average velocity , $\displaystyle = \frac{\int v dt}{\int dt}$

as u = 0 , hence

v = a t

$\displaystyle x = \frac{1}{2} a t^2 $

$\displaystyle dx = a t dt $

Space average velocity , $ = \frac{\int ( at) at dt}{\int at dt }$

$\displaystyle = \frac{\int at^2 dt}{\int t dt }$

$\displaystyle = \frac{at^3 / 3}{t^2 /2} = \frac{2}{3} a t$ …(i)

Time average velocity , $\displaystyle = \frac{\int at dt}{\int dt}$

$\displaystyle = \frac{at^2 /2}{t} = \frac{1}{2}a t $ …(ii)

$\displaystyle \frac{Space \; avg \; velocity}{Time \; avg \; velocity} = \frac{\frac{2}{3}a t}{\frac{1}{2}a t}$

$\displaystyle \frac{Space \; avg \; velocity}{Time \; avg \; velocity} = \frac{4}{3}$