A particle moves with an initial velocity v_o and retardation $\mathrm{\alpha }$ v,  where v is its velocity at any time t. The velocity of the particle at time $\mathrm{t} = 1/\mathrm{\alpha }$ will be

We know that, a = (dv / dt).
Given that $\mathrm{a}=-\mathrm{\alpha v}$
$\therefore \frac{\mathrm{dv}}{\mathrm{dt}}=-\mathrm{\alpha v} \mathrm{or} \frac{\mathrm{dv}}{\mathrm{v}}=-\mathrm{\alpha dt}$
Integrating within proper limits, we have
${\int }_{{\mathrm{V}}_0}^{\mathrm{v}} \frac{\mathrm{dv}}{\mathrm{v}}=-\mathrm{\alpha }{\int }_0^{\mathrm{t}} \mathrm{dt}$
or ${\left[{\log }_{\mathrm{e}}\mathrm{v}\right]}_{{\mathrm{v}}_0}^{\mathrm{v}}=-\mathrm{\alpha }[\mathrm{t}{]}_0^{\mathrm{t}}$
$\therefore {\log }_{\mathrm{e}}\mathrm{v}-{\log }_{\mathrm{e}}{\mathrm{v}}_0=-\mathrm{\alpha t}$
or ${\log }_{\mathrm{e}}\frac{\mathrm{V}}{{\mathrm{V}}_0}=-\mathrm{\alpha t} \mathrm{or} \frac{\mathrm{V}}{{\mathrm{V}}_0}={\mathrm{e}}^{-\mathrm{\alpha t}}$
When $\mathrm{t}=1/\mathrm{\alpha },$ we get
$\mathrm{v}={\mathrm{v}}_0{\mathrm{e}}^{-\mathrm{\alpha }\times (1/\mathrm{\alpha })}={\mathrm{v}}_0{\mathrm{e}}^{-1}={\mathrm{v}}_0/\mathrm{e}$