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Slingshot

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A slingshot is a small projectile weapon which consists of a Y-shaped frame held in one hand, with two rubber strips attached to the uprights.

Rubber strips obey Hooke's law which says that when a rubber strip is extended by a small amount $\delta x$, the force it exerts is found to be $F = k\delta x$ where $k$ is a force constant.

Suppose that you want to shoot a stone of mass $m = 50\textrm{ g}$ such that it will go to the river which is at a distance of $50$ meters. Moreover, it is given that $L = 12\textrm{ cm}$, $H = 10\textrm{ cm}$, $k = 200\textrm{ N/m}$ and $g = 9.81\textrm{ m/s}^2$.

How much do you need to extend the strips (i.e. find $x$) in order to make a shot into the river if the stone is fired at an angle of $45^\circ$ with the horizontal and the length of the unstreched strip is less than $\sqrt{L^2 - H^2}$?