Question

A particle moves with an initial velocity $v_0$ and retardation $\alpha v$, where $v$ is its initial velocity at any time $t$ and $\alpha$ is a constant

• 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 correct options are
A The particle will continue to move for a very long time.
C The particle will cover a total distance $\frac{v_0}{\alpha }$.

$\frac{dv}{dt}=-\alpha v$

$\frac{dv}{v}=-\alpha dt$

Integrating both sides, we get
$ln\left(\frac{v}{v_o}\right)=-\alpha t$

$v=v_o{e}^{-\alpha t}$
Thus, the particle will continue to move for a very long time.

To calculate the distance traveled by the particle write the differential equation as

$v\frac{dv}{ds}=-\alpha v$

${\int }_{v_o}^{0}dv=-\alpha {\int }_0^{s}ds$

Also total distance traveled will area under curve which is equal to $=\frac{v_0}{\alpha }$