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henryc09

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## Homework Statement

A hohmann transfer orbit is a way of transferring a spacecraft between two planetary orbits (which we shall assume to be circular) by using one half of an elliptical orbit about the sun.

Suppose the spacecraft is initially moving around the sun with orbital speed v1 of the first planet, at radius R1, and we wish to move it to a larger orbital radius R2. Let their orbital speeds of the spacecraft at perihelion (point A)(sorry I can't show you the diagram) and aphelion (point B) in the elliptical orbit be vA and vB respectively. Write down the conditions on vA and vB assuming the planets have a negligible gravitational effect compared to the sun based on: (i) the conservation of energy (ii) the conservation of angular momentum.

Hence show that that the velocity boost required to accelerate the spacecraft into the transfer orbit is

change in v = vA-v1 = sqrt(GM/R1) [sqrt(2R2/(R1+R2)) - 1]

where M is mass of sun

## Homework Equations

## The Attempt at a Solution

well I think that conservation of energy would be:

0.5mvA^2 - GmM/R1 = 0.5mvB^2 - GmM/R2

divide through by m

conservation of angular momentum would be:

-mR1vA = -mR2vB

R1vA=R2vB

Then you have that the centripetal force acting on the spacecraft when it's moving in a circular orbit is mv1^2/R1 = GmM/R1^2

giving v1 as sqrt(GM/R1)

but then when I say that Vb = VaR1/R2 and substitute this into the conservation of energy equation I end up with negative square roots and can't get to the result it gives. I'm probably being really stupid but could someone talk me through the steps or point out what I'm doing wrong?