Question

A particle of mass $m$ is moving in a straight line with momentum $p$. Starting at time $t=0$, a force $F=kt$ ($k$ is a constant) acts along along the direction of motion of particle for the time interval $T$ so that its momentum changes from $p$ to $5p$. The value of $T$ is

• A. $\sqrt{\frac{2p}{k}}$
• B. $2\sqrt{\frac{p}{k}}$
• C. $2\sqrt{\frac{2p}{k}}$
• D. $2\sqrt{\frac{k}{p}}$

Answer 1

The correct option is C $2\sqrt{\frac{2p}{k}}$

From newton's $2^{\text{nd}}$ of motion :
$F=\frac{dp}{dt}...\left(i\right)$
$\because F=kt,P_i=p,P_f=5p$
And $t_i=0,t_f=T$
$\Rightarrow$Integrating Eq. $\left(i\right)$ on both sides with suitable limits :
${\int }_{t=0}^{t=T}Fdt={\int }_p^{5p}dp$
${\int }_{t=0}^{t=T}ktdt={\int }_p^{5p}dp$
$\Rightarrow {\left[k\frac{t^2}{2}\right]}_0^T={\left[p\right]}_p^{5p}$
$\Rightarrow \frac{k{T}^2}{2}=[5p-p]=4p$
$\therefore T=\sqrt{\frac{8p}{k}}=2\sqrt{\frac{2p}{k}}$