First slide

Projectile motion

Question

The range R of projectile is same when its maximum heights arc h₁ and h₂. what is the relation between R, h₁, and h₂?

Moderate

A

$R = \sqrt{h_{1} h_{2}}$
B

$R = \sqrt{2 h_{1} h_{2}}$
C

$R = 2 \sqrt{h_{1} h_{2}}$
D

$R = 4 \sqrt{h_{1} h_{2}}$

Solution

Range is same for angles of projection $θ and 90^{\circ} - θ$
$\begin{array}{l} R = \frac{u^{2} \sin^{2} θ}{g}, h_{1} = \frac{u^{2} \sin^{2} θ}{2 g} \\ and h_{2} = \frac{u^{2} \sin^{2} (90 - θ)}{2 g} = \frac{u^{2} \cos^{2} θ}{2 g} \\ Hence, \sqrt{h_{1} h_{2}} = \frac{u^{2} \sin θ \cos θ}{2 g} = \frac{1}{4} [\frac{u^{2} \sin 2 θ}{g}] = \frac{R}{4} \end{array}$

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