Help! Physics derivations.

[Kurt_]

Consider a satellite of mass m moving in a circular orbit around the Earth at a constant speed v at an altitude h above the Earth’s surface.
a) Determine the speed of the satellite, v, in terms of G, h, RE (the radius of Earth), and ME (the mass of Earth).

b) Determine the satellite’s period of revolution, T (the time for one revolution about the Earth).

c) Square both sides of the result that you got in part b). Research the significance of this proportionality relation to our understanding of astronomy. Write a summary of your research. It must be less than 100 words.


Help me do this, and I'll give you a cookie. And SS will give you an omihug or whatever.

I'm pretty sure the answer to B is T^2/R^3 = something. I don't remember the exact equation.

[Black Six]

a) Law of Universal Gravitation F = (G * m1 * m2) / r^2, where m1 is the mass of the earth and r = h + RE
=> F = m2 * a = (G * m1 * m2) / r^2
=> a = (G * m1) / r^2
Since you know it's a circular orbit, you have F = (m * v^2) / r
=> m * a = (m * v^2) / r
=> a = v^2 / r
=> v^2 / r = (G * m1) / r^2
=> v^2 = (G * m1) / r
=> v = ((G * m1) / r)^(1/2)

I'd do the others, but it's time for sleep. Hope that helps.

Edit: Good morning, here's b)
The object moves with a constant velocity, therefore you can use s = v * t
For t = T, one period, the satellite travels once through it's orbit, ie the circumference of a circle with radius r (as defined earlier). So we have s = 2 * pi * r. Substituting we get:

2 * pi * r = ((G * m1) / r)^(1/2) * T
=> T = (r / (G * m1))^(1/2) * (2 * pi * r) = (2 * pi) * (r^3 / (G * m1))^(1/2)

As far as the answer you suggested for b, we can rewrite the above like this:
T^2 = 4 * (pi^2) * (r^3 / (G * m1))
=> T^2 / r^3 = (4 * pi^2) / (G * m1)

As far as c) goes, the result above is called Kepler's Third Law, which I'll let Wikipedia explain.

And remember kids, physics is fun!

[Kurt_]

I love you.