Angle of projection for the minimum time of flight for a given range

Question

If a projectile has to cover a fixed range under gravity, then what should be the angle of projection for the total time of flight to be minimum? The initial and final point of the projectile are both on level ground, gravity is constant, and the projectile is a point object. The initial velocity of the projectile is u

Answer 1

Lets look at the equations

$$x=v_{{0}}\cos \left( \varphi \right) t\tag 1$$ $$y=v_{{0}}\sin \left( \varphi \right) t-\frac 12\,g{t}^{2}\tag 2$$

solve those equations with $~x=L~$ and $~y=0~$ for $~v_0~,\varphi~$ you obtain:

$$\varphi_L=\arctan\left(\frac{g\,t_L^2}{2\,L}\right)$$ $$v_{0L}=\frac 12\frac{\sqrt{g^2\,t_L^4+4\,L^2}}{t_L}$$

hence: the angle of projection $ ~\varphi_L~$ and the start velocity $~v_{0L}~$ both are function of the time $~t_L~$ so I don't think there is unique solution for your question ?

Answer 2

Do you need to hit the final target? If so, and if your rifle is powerful enough to reach tour target, you usually get two solutions, for two different angles of incidence w.r.t. the horizontal ground (with the only exception of the target placed at the maximum range of the rifle where you get only one value of the angle).

Among these two solutions, you need to take the one with the smaller angle, an thus the maximum horizontal component of the velocity (constant throughout the motion of the projectile, if air drag is neglected), and thus the minimum time needed to cover the horizontal distance.