Question

A long time ago in a galaxy far far away there was a planet that had two moons: Dawsonia and Panaside.
Panaside was located in a stable circular orbit 2398 km from the surface of the planet, had an orbital
period of 19 hours, and was a magnificent moon which served as the intellectual center of the whole
sector. The other moon - Dawsonia - was a considerably smaller body which had no intelligent life.
Dawsonia was in a stable circular orbit 59850 km from the surface of the planet and had an orbital period
of 47 hours. Based on the orbital data of these two moons, how long will it take an object to reach the
ground if it was dropped from a height of 15 m above the surface of the planet?

Answer 1

First, here's the derivation for time period of a moon orbiting a planet.

Now, applying the above formula in given data, we obtain two equations

We proceed to solve these equations and find r_planet and M_planet

Here, we're using an approximation that g doesn't significantly vary near the surface of the planet. With a difference of 15 meters in height above the surface, the value of g changes in the order of 10^-8 m/s² in this case. So, it's reasonable to assume g is constant.

Next, the equation of motion, s=ut+at²/2 is used to find the time for an object to fall from 15 m above the surface of the planet.