Projectile range ($R$)

The horizontal displacement between launch and landing in a projectile motion model. With equal launch and landing heights, uniform gravitational acceleration $g$, no air resistance, launch speed $u$ and angle $\theta$, it is $R=u^2\sin (2\theta )/g$. Different landing heights or forces change the result.

Under uniform gravity without drag, a projectile launched at speed $u$ from height $h$ reaches a maximum horizontal projectile range $u\sqrt{u^2+2gh}/g$. For horizontal position $X$, maximize $y=h+Xz-g{X}^2(1+z^2)/(2u^2)$ over $z=\tan \alpha$. The maximizer is $z=u^2/(gX)$ and its envelope meets the ground at the displayed distance. The optimum angle obeys $\tan \alpha =u/\sqrt{u^2+2gh}$.

A projectile launched from and returning to the floor beneath a horizontal ceiling of height $H$ has maximum projectile range $\sqrt{8H(u^2/g-2H)}$ when $0\le H\le u^2/(4g)$. For higher ceilings the maximum is $u^2/g$. This assumes uniform gravitational acceleration, no air resistance and fixed launch speed $u$.

Indeed, with $z={\sin }^2\theta$, the apex condition is $z\le 2gH/u^2$ and $R^2=4u^{4}z(1-z)/g^2$. This concave quadratic function is maximized at the admissible value nearest $1/2$. Thus the lower-ceiling expression must not be extrapolated past its regime. If touching the ceiling is excluded, the limiting value is a supremum instead of an attained maximum.