A body of mass moves in the gravitational field of a much larger spherical object of mass located at the origin. Starting from the equations of motion
(i) the body moves in an orbit of the form
where is the constant angular momentum per unit mass, and and are constants;
(ii) the total energy of the body is
A meteorite is moving very far from the Earth with speed , and in the absence of the effect of the Earth's gravitational field would miss the Earth by a shortest distance (measured from the Earth's centre). Show that in the subsequent motion
Use equation to find the distance of closest approach, and show that the meteorite will collide with the Earth if
where is the radius of the Earth.