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

The correct option is C $\frac{4}{3}$

Space average velocity is $\frac{\int vdx}{\int dx}...........\left(a\right)$

Time – average velocity is $\frac{\int vdt}{\int dt}...........\left(b\right)$

Initially the particle is at rest so u = 0.

$v=at.........\left(1\right)$

$x=\frac{1}{2}a{t}^2........\left(2\right)$

$dx=atdt..............\left(3\right)$

For space-average velocity put (1) and (3) in

$\frac{\int a^2{t}^{2}dt}{\int atdt}=\frac{\frac{a{t}^3}{3}}{\frac{t^2}{2}}=\frac{2}{3}at.........\left(4\right)$

For time average velocity $\frac{\int atdt}{\int dt}=\frac{a{t}^2}{2t}=\frac{at}{2}...........\left(5\right)$

Taking ratios of (4) and (5)

$\frac{\frac{2}{3}at}{\frac{at}{2}}=\frac{4}{3}$

The ratio of space- average velocity to the time average velocity is $4:3$