Paper 4, Section II, A

In an alien invasion, a flying saucer hovers at a fixed point $S$, a height $l$ far above the White House, which is at point $W$. A wrecking ball of mass $m$ is attached to one end of a light inextensible rod, also of length $l$. The other end of the rod is attached to the flying saucer. The wrecking ball is initially at rest at point $B$, and the angle $W S B$ is $\theta_{0}$. At $W$, the acceleration due to gravity is $g$. Assume that the rotation of the Earth can be neglected and that the only force acting is Earth's gravity.

(a) Under the approximations that gravity acts everywhere parallel to the line $S W$ and that the acceleration due to Earth's gravity is constant throughout the space through which the wrecking ball is travelling, find the speed $v_{1}$ with which the wrecking ball hits the White House, in terms of the constants introduced above.

(b) Taking into account the fact that gravity is non-uniform and acts toward the centre of the Earth, find the speed $v_{2}$ with which the wrecking ball hits the White House in terms of the constants introduced above and $R$, where $R$ is the radius of the Earth, which you may assume is exactly spherical.

(c) Finally, show that

$v_{2}=v_{1}\left(1+\left(A+B \cos \theta_{0}\right) \frac{l}{R}+O\left(\frac{l^{2}}{R^{2}}\right)\right)$

where $A$ and $B$ are constants, which you should determine.

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