Air Friction Stops my Projectile

From the origin of a Cartesian coordinate system, a particle of mass mm is thrown with an initial speed v0v_{0} at an angle of θ=π3\theta=\frac{\pi}{3} from the horizontal x-axis. The air friction ff acts on the particle such that f=bv,\vec{f} = - b\vec{v}, where bb is a positive constant and v\vec{v} is the velocity of the particle. If the x cordinate where the particle will only have vertical speed can be writen as x=αmv0βb.x=\dfrac{\alpha\cdot mv_{0}}{\beta \cdot b}. Find the value of α+β.\alpha+\beta.

The gravity acts in the vertically downward direction as g=gy^\vec{g}=-g\hat{y}.


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