# A particle of mass M originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation

Question:

A particle of mass M originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation

$\mathrm{F}=\mathrm{F}_{0}\left[1-\left(\frac{\mathrm{t}-\mathrm{T}}{\mathrm{T}}\right)^{2}\right]$

Where $\mathrm{F}_{0}$ and $\mathrm{T}$ are constants. The force acts only for the time interval $2 \mathrm{~T}$. The velocity $\mathrm{v}$ of the particle after time $2 \mathrm{~T}$ is :

1. $2 \mathrm{~F}_{0} \mathrm{~T} / \mathrm{M}$

2. $\mathrm{F}_{0} \mathrm{~T} / 2 \mathrm{M}$$4 \mathrm{~F}_{0} \mathrm{~T} / 3 \mathrm{M}$

3. $4 \mathrm{~F}_{0} \mathrm{~T} / 3 \mathrm{M}$

4. $\mathrm{F}_{0} \mathrm{~T} / 3 \mathrm{M}$

Correct Option: , 3

Solution:

$\mathrm{t}=0, \mathrm{u}=0$

$\mathrm{a}=\frac{\mathrm{F}_{\mathrm{o}}}{\mathrm{M}}-\frac{\mathrm{F}_{\mathrm{o}}}{\mathrm{MT}^{2}}(\mathrm{t}-\mathrm{T})^{2}=\frac{\mathrm{dv}}{\mathrm{dt}}$

$\int_{0}^{\mathrm{v}} \mathrm{dv}=\int_{\mathrm{t}=0}^{2 \mathrm{~T}}\left(\frac{\mathrm{F}_{\mathrm{o}}}{\mathrm{M}}-\frac{\mathrm{F}_{\mathrm{o}}}{\mathrm{MT}^{2}}(\mathrm{t}-\mathrm{T})^{2}\right) \mathrm{dt}$

$\mathrm{V}=\left[\frac{\mathrm{F}_{\mathrm{o}}}{\mathrm{M}} \mathrm{t}\right]_{0}^{2 \mathrm{~T}}-\frac{\mathrm{F}_{\mathrm{o}}}{\mathrm{MT}^{2}}\left[\frac{\mathrm{t}^{3}}{3}-\mathrm{t}^{2} \mathrm{~T}+\mathrm{T}^{2} \mathrm{t}\right]_{0}^{2 \mathrm{~T}}$

$V=\frac{4 F_{0} T}{3 M}$