An artillery gun is located on a plane surface inclined at

Question

An artillery gun is located on a plane surface inclined at an angle β to the horizontal. The gun is aligned with the line of steepest slope of the plane. The gun fires a shell with speed u in the direction making an angle α with the (upward) line of steepest slope.

Find where the shell lands. Deduce the maximum ranges R^U, R^D, up and down the plane, and show that

R^U/R^D = (1 - sin β) / (1 + sin β).