A hunter tries to hunt a monkey with a small, very poisonous arrow, blown from a pipe with initial speed v₀. The monkey is hanging on a branch of a tree at height H above the ground. The hunter is at a distance L from the bottom of the tree. The monkey sees the arrow leaving the blow pipe and immediately loses the grip on the tree, falling freely down with zero initial velocity. The minimum initial speed v₀ of the arrow for hunter to succeed while monkey is in air, is

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$\sqrt{\frac{2 {gH}^{2}}{\sqrt{H^{2} + L^{2}}}}$

$\sqrt{\frac{g (H^{2} + L^{2})}{H}}$

$\sqrt{\frac{g (H^{2} + L^{2})}{2 H}}$

$\sqrt{\frac{{gH}^{2}}{\sqrt{H^{2} + L^{2}}}}$

answer is A.

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Detailed Solution

For minimum speed, arrow will hit the monkey just at ground.

$\begin{array}{l} T = \frac{2 v_{0} \sin θ}{g} = \sqrt{\frac{2 H}{g}} \\ \Rightarrow v_{0} \sin θ = \sqrt{\frac{gH}{2}} . . . . . . . . (i) \\ R = L = (v_{0} \cos θ) \sqrt{\frac{2 H}{g}} \\ v_{0} \cos θ = \sqrt{\frac{{gL}^{2}}{2 H}} . . . . . . . . (ii) \end{array}$

Adding and squaring Eqs. (i) and (ii), we get

$v_{0} = \sqrt{\frac{g (H^{2} + L^{2})}{2 H}}$

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