A New Earth!

Classical Mechanics Level 5

A strangely spherical planet called Golplan was found and an initial test confirmed that the planet was made of homogeneous gold (WoW!). A group of scientists along with some workmen who were skilled in excavations was send in a probe to Golplan. A narrow trial shaft was bored from point A on its surface to the centre of the planet. At that point, an accident occurred when one of the workers fell off the surface of the planet into the trial shaft. He fell, speeding as he got nearer to the centre, and finally reached \(O\), where he died. However, the work was continued and the group started excavation of gold, forming a spherical cavity of diameter \(AO\) in the planet, as illustrated.

Then a second accident occurred - another workman similarly fell from the point A to point \(O\) and died. An expert scientist was asked to calculate the ratio of the impact speeds and the ratio of times taken to fall from \(A\) to \(O\) by the two unfortunate workers.

If we denote \( x = \dfrac{T_1}{T_2} \) and \(y = \dfrac{v_1}{v_2} \), find the value of \( \dfrac xy\).

Assume that the volume of the narrow trial shaft to be negligible.


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