To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height H = v 2 sin 2 ⁡ ⁡ ( θ θ ) / ( 2 g ) with respect to θ θ , that is d H d θ θ = v 2 2 cos ⁡ ⁡ ( θ θ ) sin ⁡ ⁡ ( θ θ ) / ( 2 g ) which is zero when θ θ = π π / 2 = 90 ∘ ∘ . So the maximum height H m a x = v 2 2 g is obtained when the projectile is fired straight up.