A space vehicle approaching a planet has a speed v, when it is very far from the planet. At that moment tangent of its trajectory would miss the centre of the planet by distance R. If the planet has mass M and radius r, what is the smallest value of R in order that the resulting orbit of the space vehicle will just miss the surface of the planet?

Question

A space vehicle approaching a planet has a speed v, when it is very far from the planet. At that moment tangent of its trajectory would miss the centre of the planet by distance R. If the planet has mass M and radius r, what is the smallest value of R in order that the resulting orbit of the space vehicle will just miss the surface of the planet?

Infinity Learn · Accepted Answer

From the principle of conserving angular momentum, we have mvR $$=$$ mv'r[v' $$=$$ speed when spaceship is just touching the planet] From conserving of energy, we have $$12mv2=12mv$$'$$2$$−GMmr. Solving we get $$R=rvv2+2GMr1/2$$.

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