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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
$F=F_{0}\left(1-\left(\frac{t-T}{T}\right)^{2}\right)$
Where $F_{0}$ and $T$ are constants. The force acts only for the time internal $2 T$. The velocity $v$ of the particle after time $2 {T}$ is -
$\frac{{F}_{0} {T}}{3 {M}}$
$\frac{{F}_{0} {T}}{2 {M}}$
$\frac{2{F}_{0} {T}}{{M}}$
$\frac{4 {F}_{0} {T}}{3 {M}}$
Solution
${t}=0, {u}=0$
As given
${a}=\frac{{F}_{0}}{{M}}-\frac{{F}_{{o}}}{{MT}^{2}}({t}-{T})^{2}=\frac{{dv}}{{dt}}$
$\int_{0}^{v} {d} v=\int_{t=0}^{2 T}\left(\frac{F_{o}}{M}-\frac{F_{0}}{M T^{2}}(t-T)^{2}\right) d t$
$V=\left[\frac{F_{0}}{M} t\right]_{0}^{2 T}-\frac{F_{0}}{M T^{2}}\left[\frac{t^{3}}{3}-t^{2} T+T^{2} t\right]_{0}^{2 T}$
${V}=\frac{4 {F}_{0} {T}}{3 {M}}$
