question_answer

A ball is thrown upward with an initial velocity \[{{V}_{0}}\]from the surface of the earth. The motion of the ball is affected by a drag force equal to \[m\gamma {{\upsilon }^{2}}\] (where m is mass of the ball, \[\upsilon \]is its instantaneous velocity and \[\gamma \] is a constant). Time taken by the ball to rise to its zenith is : [JEE Main 10-4-2019 Morning]

A) \[\frac{1}{\sqrt{\gamma g}}{{\sin }^{-1}}\left( \sqrt{\frac{\gamma }{g}}{{V}_{0}} \right)\]         

B) \[\frac{1}{\sqrt{\gamma g}}{{\tan }^{-1}}\left( \sqrt{\frac{\gamma }{g}}{{V}_{0}} \right)\]

C)

\[\frac{1}{\sqrt{2\gamma g}}{{\tan }^{-1}}\left( \sqrt{\frac{2\gamma }{g}}{{V}_{0}} \right)\]

D) \[\frac{1}{\sqrt{\gamma g}}\ln \left( 1+\sqrt{\frac{\gamma }{g}}{{V}_{0}} \right)\]

Correct Answer: B

Solution :

[b]             Integrating :-