# A particle of mass m, originally at rest, is subjected to a force whose direction is constant but …..

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

$\displaystyle F = F_0[ 1 – (\frac{t-T}{T})^2 ]$

where F0 and T are constants. The force acts only for the time interval 2T.

Prove that the speed v of the particle after a time 2T has elapsed is equal to $\displaystyle \frac{4F_0 T}{3 m}$

Solution :

$\displaystyle F = F_0[ 1 – (\frac{t-T}{T})^2 ]$

$\displaystyle v = \frac{F_0}{m}\int_{0}^{2T}[ 1 – (\frac{t-T}{T})^2 ]dt$

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