Derive an equation for the orbital velocity of the planets in our solar system,

Question

Derive an equation for the orbital velocity of the planets in our solar system, by using the (more) complete version of Kepler's 3rd law and assming circular orbits. Follow these steps: first find an expression for orbital period in terms of orbital velocity for a circular orbit, then subsitute this expression into Kepler's 3rd law and finally solve for orbital velocity. Clearly show your work, describe any variables used and show your derivations.

Kepler's Law : p^2 = a^3

Kepler's complete 3rd Law: GM = (4(pi^2)r^3)/ p^2

Explanation / Answer

let r is the radius of orbit and v is the orbital speed.

orbital period, T = 2*pi*r/v

2*pi*r^(3/2)sqrt(G*M) = 2*pi*r/v

sqrt(r/(G*M)) = 1/v

==> v = sqrt(G*M/r) <<<<<------------Answer

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