A particle starts from rest and accelerates uniformly. The ratio of space-average velocity to time-average velocity in the given time 't' is

$\begin{array}{l}\mathrm{V}=\mathrm{u}+\mathrm{at}\Rightarrow \mathrm{V}=\mathrm{at};\mathrm{x}=\frac{1}{2}{\mathrm{at}}^2\\ \mathrm{dx}=\frac{1}{2}\mathrm{a}\left(2\mathrm{t}\right)\mathrm{dt}=\mathrm{atdt}\end{array}$
Space ave.Velocity $\left({\mathrm{V}}_1\right)=\frac{\int \mathrm{udx}}{\int \mathrm{dx}}=\frac{\int \mathrm{at}\times \mathrm{atdt}}{\int \mathrm{atdt}}$
$=\frac{{\mathrm{a}}^2\int {\mathrm{t}}^2\mathrm{dt}}{\int \mathrm{atdt}}=\frac{{\mathrm{a}}^2\frac{{\mathrm{t}}^3}{3}}{\mathrm{a}\left(\frac{{\mathrm{t}}^2}{2}\right)}=\frac{2\mathrm{a}}{3}\times \mathrm{t}$

Time ave. Velocity
$\begin{array}{l}\left({\mathrm{V}}_2\right)=\frac{\int \mathrm{udt}}{\int \mathrm{dt}}=\frac{\int \mathrm{atdt}}{\int \mathrm{dt}}=\frac{\mathrm{a}\frac{{\mathrm{t}}^2}{2}}{\mathrm{t}}=\frac{\mathrm{at}}{2}\\ {\mathrm{V}}_1:{\mathrm{V}}_2=\frac{\frac{2}{3}\mathrm{at}}{\frac{\text{ at }}{2}}=\frac{4}{3}\end{array}$