Two stones are projected from the top of a cliff h metres high, with the same speed u, so …

Question

Two stones are projected from the top of a cliff $h$ metres high, with the same speed $u$, so as to hit the ground at the same spot. If one of the stones is projected horizontally and the other is projected at an angle $	heta$ to the horizontal, then $	an	heta$ equals:

Accepted Answer

For horizontal projection: $R=ut$, $h=\dfrac12gt^2 \implies t=\sqrt{\dfrac{2h}g} \implies R=u\sqrt{\dfrac{2h}g}$ \ldots (1)

For projection at angle $\theta$: $R=u\cos\theta\cdot t'$, $h=-u\sin\theta\cdot t'+\dfrac12gt'^2$

From (1) and the horizontal equation, $t'=\dfrac1{\cos\theta}\sqrt{\dfrac{2h}g}$. Substituting into the vertical equation and simplifying:

$$\tan^2\theta - \frac{u^2}{hg}\tan\theta = 0 \implies \tan\theta=\frac{u^2}{hg}$$

Solutions · 1