Question

prove that a gun will shoot 3 times as high as when it is fired at an angle of 60 degrees as when it is fired at an angle of 30 degrees but that the horizontal range is is the same in each.

Answer 1

$for \theta = 60 \textdegree \\ \\ \text{ Horizontal range }(h_1) = \frac{u^{2} \sin^{2}\theta}{2g} \\ \\ h_1 = \frac{u^{2} \sin^{2} 60 \textdegree}{2g} \\ \\ h_1 = \frac{u^{2}(\frac{\sqrt{3}}{2})^{2}}{2g} \\ \\ h_1 = \frac{3u^{2}}{8g} \\ \\ Range = \frac{{u^{2} \sin2\theta}}{g} \\ \\ R_1 = \frac{{u^{2} \sin2*60 \textdegree}}{g} \\ \\ R_1 = \frac{{u^{2} \sin120 \textdegree}}{g} \\ \\ R_1 = \frac{u^{2} \sin(90+30)}{g} \\ \\ R _1 = \frac{u^{2} \cos 30}{g} \\ \\ R_1 = \frac{\sqrt{3}u^{2}}{2g}$$\\ \\ for \ \theta = 30 \textdegree\text{ Horizontal range }(h_2) = \frac{u^{2} \sin^{2}\theta}{2g} \\ \\ h_2 = \frac{u^{2} \sin^{2} 30 \textdegree}{2g} \\ \\ h_2 = \frac{u^{2}(\frac{\1}{2})^{2}}{2g} \\ \\ h_2 = \frac{u^{2}}{8g} \\ \\ Range = \frac{{u^{2} \sin2\theta}}{g} \\ \\ R_2 = \frac{{u^{2} \sin2*30 \textdegree}}{g} \\ \\ R_2 = \frac{{u^{2} \sin60 \textdegree}}{g} \\ \\ R _2 = \frac{u^{2} \sin 60}{g} \\ \\ R_2 = \frac{\sqrt{3}u^{2}}{2g}$$\textbf{Hence,} \\ \\ \frac{h_1}{h_2} = \frac{3}{1} \\ \\ h_1 = 3h_2 \\ \\ R_1 = R_2 \\ \\ \textbf{Proved}$