The Enterprise wants to orbit a 4.15 x 1024 kg planet with a period of 14100 s. What should the radius of their orbit be?

Answer 1

Hi Tamia!

This problem is a classic uniform circular motion problem. We want to figure out how orbital period is related to orbital radius and the mass of the planet being orbited.

To do so, the critical realization is that the gravitational force between the planet and the Enterprise provides the centripetal force for the orbital motion. That is,

GM_planetM_enterprise/r² = M_enterprisev²/r.

Note that the mass of the enterprise cancels on both sides, so that we get

GM_planet/r² = v²/r.

This allows us to solve for the velocity of the planet as a function of the orbital radius:

v = √(GM_planet/r).

Now, how does this help us find the orbital period? Well, the period is the time it takes the enterprise to complete a full orbit. The distance covered in a full orbit is 2πr meters, since it's simply the circumference of a circle with radius r!

Using our usual equation for distance as a function of time (x = vt), this tells us that

2πr = √(GM_planet/r)* t.

This completes most of the work for your problem. You know the values for r, G, and t, the orbital period. Plug those in, and solve for M!

Hope this was helpful :)