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?

A)  

B)  

C)  

D)  

Correct Answer: C

Solution :

[c] \[F=ma={{F}_{0}}{{e}^{-bt}}\] \[\Rightarrow \frac{dv}{dt}=\frac{{{F}_{0}}}{m}{{e}^{-bt}}\] \[\Rightarrow \int\limits_{0}^{v}{dv=\frac{{{F}_{0}}}{m}\int\limits_{0}^{t}{{{e}^{-bt}}dt}}\] \[\Rightarrow v=\frac{{{F}_{0}}}{m}{{\left[ \frac{{{e}^{-bt}}}{-b} \right]}_{0}}^{t}\] \[\Rightarrow v=\frac{{{F}_{0}}}{mb}(1-{{e}^{-bt}})\]