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 `mgammav^2` (where m is mass of the ball, v is its instantaeous velocity and `gamma` is a constant).Time taken by the ball to rise to its zenith is :

Options

A. `1/sqrt(gammag)"sin"^(-1) (sqrt(gamma/g)V_0)`
B. `1/sqrt(gammag)"tan"^(-1) (sqrt(gamma/g)V_0)`
C. `1/sqrt(2gammag)"tan"^(-1) (sqrt((2gamma)/g)V_0)`
D. `1/sqrt(gammag)ln^(-1) (1+sqrt(gamma/g)V_0)`

Answer

B

Explanation

`a=(dv)/(dt)=-(g+gammav^2)rArr int_(v_0)^0 (dv)/((g+gammav^2))=int_0^t dt rArr -1/gamma int_(v_0)^0 (dv)/(v^2+(sqrt(g//r)^2))=int_0^t dt`
`rArr t=1/gamma 1/(sqrt(g//r)) [tan^(-1) [v/sqrt(g//r)]]_0^(v_0)` or `t=sqrt(1/(ggamma)) tan^(-1) [sqrt(gamma/g)]`