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 m

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 ? □

TestHub · Accepted Answer

Far from the planet the vehicle has speed v and impact parameter R, so its angular momentum about the planet's centre is $L = m v R$ .

For the orbit to just miss the surface, the closest approach is the planet's radius r, where the velocity v' is perpendicular to the radius vector. Angular momentum conservation: $m v R = m v^{'} r$ .

Energy conservation between infinity and closest approach: $\frac{1}{2} m v^{2} = \frac{1}{2} m v^{' 2} - \frac{GM m}{r}$ , so $v^{' 2} = v^{2} + \frac{2 GM}{r}$ .

Hence $R = \frac{r v ^{'}}{v} = \frac{r}{v} [v^{2} + \frac{2 GM}{r}]^{1/2}$ .

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