Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2019

In an alien invasion, a flying saucer hovers at a fixed point SS, a height ll far above the White House, which is at point WW. A wrecking ball of mass mm is attached to one end of a light inextensible rod, also of length ll. The other end of the rod is attached to the flying saucer. The wrecking ball is initially at rest at point BB, and the angle WSBW S B is θ0\theta_{0}. At WW, the acceleration due to gravity is gg. 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 SWS 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 v1v_{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 v2v_{2} with which the wrecking ball hits the White House in terms of the constants introduced above and RR, where RR is the radius of the Earth, which you may assume is exactly spherical.

(c) Finally, show that

v2=v1(1+(A+Bcosθ0)lR+O(l2R2))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 AA and BB are constants, which you should determine.

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