First slide

Projectile motion

Question

Canon A is located on a plain a distance L from a wall of height H. On top of this wall is an identical cannon (cannon B). Ignore air resistance throughout this problem. Also ignore the size of the cannons relative to L and H. The two groups of gunners aim the cannons directly at each other. They fire at each other simultaneously, with equal muzzle speed v₀. What is the value of v₀ for which the two cannon balls collide just as they hit the ground?

Difficult

A

$\frac{1}{2} \sqrt{\frac{g (L^{2} + H^{2})}{H}}$
B

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

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

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

Solution

$\begin{array}{l} v_{0} \cos θ (\frac{2 v_{0} \sin θ}{g}) + v_{0} \cos θ (\frac{2 v_{0} \sin θ}{g}) = L \\ \sin θ = \frac{H}{\sqrt{H^{2} + L^{2}}} and \cos θ = \frac{L}{\sqrt{H^{2} + L^{2}}} \\ v_{0}^{2} = \frac{g L}{4 \sin θ \cos θ} \\ \Rightarrow v_{0} = \sqrt{\frac{g (L^{2} + H^{2})}{4 H}} = \frac{1}{2} \sqrt{\frac{g (L^{2} + H^{2})}{H}} \end{array}$

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