Dynamics | Part IA, 2001

A spacecraft of mass mm moves under the gravitational influence of the Sun of mass MM and with universal gravitation constant GG. After a disastrous manoeuvre, the unfortunate spacecraft finds itself exactly in a parabolic orbit about the Sun: the orbit with zero total energy. Using the conservation of energy and angular momentum, or otherwise, show that in the subsequent motion the distance of the spacecraft from the Sun r(t)r(t) satisfies

(rr0)(r+2r0)2=92GM(tt0)2,\left(r-r_{0}\right)\left(r+2 r_{0}\right)^{2}=\frac{9}{2} G M\left(t-t_{0}\right)^{2},

with constants r0r_{0} and t0t_{0}.

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