Air Friction Stops my Projectile

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

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


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