Velocity of earths orbit?

1. The problem statement, all variables and given/known data
Given that the earth’s distance from the sun varies from 1.47 to 1.52×10^11m, determine the minimum and maximum velocities of the Earth in it’s orbit around the sun.

2. Relevant equations

[itex] F=G\frac{m1m2}{r^2} [/itex]

[itex] E=K+U [/itex] ???

3. The attempt at a solution

I think the way to do this is with K1+U1 = K2+U2 , where one side of the equation is the earth at its closest point to the sun and the other side is the earth at its farthest point. Let Me = mass of earth, Ms = mass of sun, Rn = distance at nearest point, Rf= distance at farthest point, Vn = velocity at nearest point, Vf = velocity at farthest point.

[itex] K1+U1 = K2 + U2 [/itex]

[itex] \frac{MeVn^2}{2} + G\frac{MsMe}{Rn} =\frac{MeVf^2}{2} + G\frac{MsMe}{Rf} [/itex]

the Me’s cancel. to solve for Vn replace Vf with [itex] \frac{2piRf}{T} [/itex]

[itex] \frac{Vn^2}{2} + G\frac{Ms}{Rn} =\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf} [/itex]

[itex] \frac{Vn^2}{2} =\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf} – G\frac{Ms}{Rn} [/itex]

[itex] Vn^2 = 2(\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf} – G\frac{Ms}{Rn}) [/itex]

[itex] Vn = (2(\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf} – G\frac{Ms}{Rn}))^\frac{1}{2} [/itex]

then after plugging in I would go back and solve for Vf. Would this give me the correct answer?

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