A particle is projected vertically upwards at speed from the surface of the Earth, which may be treated as a perfect sphere. The variation of gravity with height should not be ignored, but the rotation of the Earth should be. Show that the height of the particle obeys
where is the radius of the Earth and is the acceleration due to gravity measured at the Earth's surface.
Using dimensional analysis, show that the maximum height of the particle and the time taken to reach that height are given by
where and are functions of .
Write down the equation of conservation of energy and deduce that
Hence or otherwise show that