A particle moves with an initial velocity vo and retardation α v,  where v is its velocity at any time t. The velocity of the particle at time t = 1/α will be

We know that, a = (dv / dt).Given that a=−αv∴dvdt=−αv or dvv=−αdtIntegrating within proper limits, we have∫V0v dvv=−α∫0t dtor loge⁡vv0v=−α[t]0t∴loge⁡v−loge⁡v0=−αtor loge⁡VV0=−αt or VV0=e−αtWhen t=1/α, we getv=v0e−α×(1/α)=v0e−1=v0/e