Question

The range R of projectile is same when its maximum heights are h₁ and h₂. What is the relation between R and h₁ & h₂ ?

Moderate

Solution

we know that range is same for angle of projection $θ$ and $(90^{\circ} - θ) .$ The range and maximum heights are given by
$R = \frac{u^{2} \sin 2 θ}{g}$
$\begin{array}{l} h_{1} = \frac{(usin θ)^{2}}{2 g} and h_{2} = \frac{{usin (90 - θ)}^{2}}{2 g} = \frac{(ucos θ)^{2}}{2 \bar{g}} \\ ∴ h_{1} h_{2} = \frac{u^{2} \sin^{2} θ}{2 g} \times \frac{u^{2} \cos^{2} θ}{2 g} \\ or \sqrt{(h_{1} h_{2})} = \frac{u^{2}}{2 g} \times \sin θcos θ = \frac{1}{4} [\frac{u^{2} \sin 2 θ}{g}] \\ or \sqrt{(h_{1} h_{2})} = \frac{R}{4} or R = 4 \sqrt{(h_{1} h_{2})} \end{array}$

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