question_answer

A particle of mass m is at rest at the origin at time\[t=0\]. It is subjected to a force \[F(t)={{F}_{0}}{{e}^{-bt}}\] in the \[x\] direction. Its speed v(t) is depicted by which of the following curves? [AIEEE 2012]

A)

B)

C)  

D)

Correct Answer: C

Solution :

[c] \[F=ma={{F}_{0}}{{e}^{-bt}}\]

\[\frac{dv}{dt}=\frac{{{F}_{0}}}{m}{{e}^{-bt}}\]

\[\int\limits_{0}^{v}{dv}=\frac{{{F}_{0}}}{m}\int\limits_{0}^{t}{{{e}^{-bt}}dt}\]

\[v=\frac{{{F}_{0}}}{m}\left[ \frac{{{e}^{-bt}}}{-b} \right]_{0}^{t}\]

\[v=\frac{{{F}_{0}}}{mb}\left( 1-{{e}^{-bt}} \right)\]