A group of astronauts in a spaceship are attempting to land on Mars. As they approach the planet, they begin to plan their descent to the surface.

Question

A group of astronauts in a spaceship are attempting to land on Mars. As they approach the planet, they begin to plan their descent to the surface.(a) Before the spaceship can land on Mars, it must enter a stable “parking orbit” above the planet. Derive an equation to find the orbital speed of the rocket (with a mass $ m $), at a height $ h $ above the surface of Mars (with a mass $ M $ and a radius $ R $).
, (b) Two of the astronauts are trying to determine $ g $ at height $ h $ above Mars. Astronaut A says that $ g $ depends only on the mass of the spaceship, while Astronaut B argues that it depends only on the orbital radius, $ r = R + h $. Which aspects of each astronaut’s reasoning are correct AND incorrect? Explain your reasoning for both.
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Rita · Accepted Answer

$\boxed{v_x = \sqrt{\frac{GM}{R+h}}}$.
Gravitational acceleration at height $h$ is given by $g = \frac{GM}{(R+h)^2}$. Astronaut A is incorrect because $g$ does not depend on the spaceship&#8217;s mass, while Astronaut B is correct that $g$ depends on the orbital radius $R+h$, but his reasoning is incomplete without noting the dependence on Mars&#8217; mass.

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