question_answer

Which of the following option correctly describes the variation of the speed v and acceleration 'a' of a point mass falling vertically in a viscous medium that applies a force F = -kv, where 'k' is constant , on the body ? (Graphs are schematic and not drawn to scale)   JEE Main Online Paper (Held On 09 April 2016)

A)                                                    

B)

C)                

D)

Correct Answer: D

Solution :

                \[a=g-\alpha v\]                 \[\frac{dv}{dt}=g-\alpha v\]                 \[\int\limits_{0}^{v}{\frac{dv}{g-\alpha v}}=\int\limits_{0}^{t}{dt}\Rightarrow \ell n\left( \frac{g-\alpha v}{v} \right)=-\alpha t\]                 \[V={{v}_{0}}(1-{{e}^{-\alpha t}}),a=\frac{dv}{dt}={{v}_{0}}\alpha {{e}^{-\alpha t}}={{a}_{0}}{{e}^{-\alpha t}}\]