Question

A particle of mass $m$ is acted upon by a force $F$ given by the empirical law $F = \frac{R}{t^2} v(t).$
If this law is to be tested experimentally by observing the motion starting from rest, the best way is to plot :

• A. v(t) ageints $t^2$
• B. log v(t) against $\frac{1}{t^2}$
• C. log v(t) against $t$
• D. log v(t) against $\frac{1}{t}$

Answer 1

Answer: (D)

$m \frac{ d V }{ dt }=\frac{ R }{ t ^{2}} V$
$\Rightarrow m \frac{d v}{v}=R \frac{d t}{t^{2}}$
$\Rightarrow m \int\limits_{V_{1}}^{v_{2}} \frac{d V}{V}=R \int\limits_{t_{1}}^{t_{2}} \frac{d t}{t^{2}} $
$\Rightarrow m \ell n\left(\frac{V_{2}}{V_{1}}\right)=\left.\frac{-R}{t}\right|_{t_{1}} ^{t_{2}} $
$\Rightarrow \ln \left(\frac{V_{2}}{V_{1}}\right)=\frac{-R}{m}\left(\frac{1}{t_{2}}-\frac{1}{t_{1}}\right)$
$log\,V \,vs \frac{1}{t}$ will be a st. line curve