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)

$m\frac{dV}{dt}=\frac{R}{t^2}V$
$\Rightarrow m\frac{dv}{v}=R\frac{dt}{t^2}$
$\Rightarrow m\underset{V_1}{\overset{v_2}{\int }}\frac{dV}{V}=R\underset{t_1}{\overset{t_2}{\int }}\frac{dt}{t^2}$
$\Rightarrow m\ell n\left(\frac{V_2}{V_1}\right)={\frac{-R}{t}|}_{t_1}^{t_2}$
$\Rightarrow \ln \left(\frac{V_2}{V_1}\right)=\frac{-R}{m}\left(\frac{1}{t_2}-\frac{1}{t_1}\right)$
$logVvs\frac{1}{t}$ will be a st. line curve