The displacement $$x$$ of a particle varies with time $$t$$ as $$x = a{e^{ - \alpha t}} + b{e^{\beta t}},$$    where $$a,b,\alpha $$  and $$\beta $$ are positive constants. The velocity of the particle will

Solution :
Given, $$x = a{e^{ - \alpha t}} + b{e^{\beta t}}$$
Velocity $$v = \frac{{dx}}{{dt}} = - a\alpha {e^{ - \alpha t}} + b\beta {e^{\beta t}} = A + B$$
where, $$A = - a\alpha {e^{ - \alpha t}}$$
$$B = b\beta {e^{\beta t}}$$
The value of term $$A = - a\alpha {e^{ - \alpha t}}$$   decreases and of term $$B = b\beta {e^{\beta t}}$$   increases with time. As a result, velocity goes on increasing with time.