Chapter 4, Problem 80E

If a projectile is fired with an initial velocity v at an angle of inclination θ from the horizontal, then its trajectory, neglecting air resistance, is the parabola

$y=(\tan \theta )x-\frac{g}{2v^2{\cos }^2\theta }{x}^{2}0<\theta <\frac{\pi }{2}$

(a) Suppose the projectile is fired from the base of a plane that is inclined at an angle α, α > 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by

$R(\theta )=\frac{2v^2\cos \theta \sin (\theta -\alpha )}{g{\cos }^2\alpha }$

(b) Determine θ so that R is a maximum,

(c) Suppose the plane is at an angle α below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R.