Question:

A ball is thrown upward with an initial velocity V₀ from the surface of the earth. The motion of the ball is affected by a drag force equal to mγv²(where m is mass of the ball, v is its instantaneous velocity and γ is a constant). Time taken by the ball to rise to its zenith is:

1√γgtan⁻¹(√(γg)V₀)

1√γgln(1+√(γg)V₀)

1√γgsin⁻¹(√(γg)V₀)

1√2γgtan⁻¹(√(2γg)V₀)

-(g+γv²)=dv/dt
-gdt = dv/(gγ+v²)
Integrating 0→t V₀→0:
-gt = -√(gγ)tan⁻¹(v/√(gγ))
t = 1√γgtan⁻¹(√(γg)V₀)