Question

A particle of mass $m$ starting from rest moves along a straight line under the action of force $f$ varying with time as $t=t_0[1-\left({\frac{t-T}{T}}^2\right)]$ where $f_0$ and $T$ are positive constant.

• A. The speed of particle after a time $2T$ is $\frac{4f_{0}T}{3m}$
• B. After time $3T$, the particle starts moving backwards
• C. Between time instants $O$ and $2T$, the acceleration first increases and then decreases.
• D. The speed of particle after a time $2T$ is $\frac{5f_{0}T}{3m}$

Answer 1

The correct option is A The speed of particle after a time $2T$ is $\frac{4f_{0}T}{3m}$

$\begin{array}{c}\text{Correct option is}\left(A\right)\\ \begin{array}{cc}F& =t_0[1-\left(\frac{t-T^2}{T}\right)]\text{(Given)}\\ \frac{mdv}{dt}=t_0-\frac{t_0}{T}(t-T^2)& \end{array}\\ {\int }_0^{v}mdv={\int }_0^{2T}{t}_{0}dt-\frac{t_0}{T}(t-T^2)dt]\end{array}$

$\begin{array}{cc}{mv|}_0^v& ={[t_{0}t-\frac{t_0{t}^2}{2T}+t_{0}Tt]}_0^{2T}\\ mv& =2t_{0}T-2t_{0}T+\frac{4t_{0}T}{3}\\ v& =\frac{4t_{0}T}{3m}\end{array}$