A particle moves in the gravitational field of the Sun. The angular momentum per unit mass of the particle is and the mass of the Sun is . Assuming that the particle moves in a plane, write down the equations of motion in polar coordinates, and derive the equation
where and .
Write down the equation of the orbit ( as a function of ), given that the particle moves with the escape velocity and is at the perihelion of its orbit, a distance from the Sun, when . Show that
and hence that the particle reaches a distance from the Sun at time .