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

$F=G\frac{m1m2}{r^2}$

$E=K+U$ ???

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.

$K1+U1 = K2 + U2$

$\frac{MeVn^2}{2} + G\frac{MsMe}{Rn} =\frac{MeVf^2}{2} + G\frac{MsMe}{Rf}$

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

$\frac{Vn^2}{2} + G\frac{Ms}{Rn} =\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf}$

$\frac{Vn^2}{2} =\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf} – G\frac{Ms}{Rn}$

$Vn^2 = 2(\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf} – G\frac{Ms}{Rn})$

$Vn = (2(\frac{2pi^2Rf^2}{T^2} + G\frac{Ms}{Rf} – G\frac{Ms}{Rn}))^\frac{1}{2}$

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

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