Question

A particle moves with an initial velocity $v_0$ and retardation $\alpha v$, where $v$ is it’s velocity at any time $t$

• A. The particle will cover a total distance $\frac{v_0}{\alpha }$
• B. The particle will come to rest after a time $\frac{1}{\alpha }$
• C. The particle will continue to move for a very long time
• D. The velocity of the particle will become $\frac{v_0}{2}$ after a time $\frac{1}{\alpha }$

Answer 1

Answer: (A), (C)

The particle will continue to move for a very long time

$v\frac{dv}{dx}=-\alpha v$
or $\underset{v_0}{\overset{0}{\int }}dv=-\alpha \underset{0}{\overset{x_0}{\int }}dx$
or $x_0=\frac{v_0}{\alpha }$
$\frac{dv}{dt}=-\alpha v$ or $\underset{v_0}{\overset{v}{\int }}\frac{dv}{v}=-\alpha \underset{0}{\overset{t}{\int }}dt$
or $v=v_0{e}^{-\alpha t}$
$\therefore v=0$ for $t\to \infty$

If we put $v=\frac{v_0}{2}$
We get time as $t=\frac{ln2}{\alpha }$