Angular momentum problem - what am I doing wrong?

[henryc09]

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?

 

[losersia]

r u frm UCL??

 

[kerimek]

As a scientist, it is important to remember that making mistakes and encountering difficulties is a natural part of the learning process. In this case, it seems like you may have made a small error in your calculations. It is always a good idea to double check your work and make sure all of your equations and variables are correct before moving on to the next step.

One potential issue could be with your equation for conservation of angular momentum. Remember that the angular momentum of an object in orbit is equal to its mass times its velocity times the distance from the center of mass. In this case, the distance from the center of mass is not just R1 or R2, but the distance between the two points A and B on the elliptical orbit. So, your equation should be -mR1vA = -m(R2-R1)vB, which will give you a different relationship between vA and vB.

Another thing to keep in mind is that when working with orbital mechanics, it is important to use the correct values for the gravitational constant G and the masses of the objects involved. Make sure you are using the correct values for the sun and the spacecraft in your calculations.

Overall, it seems like you are on the right track and have a good understanding of the concepts involved. Just double check your equations and calculations, and don't be afraid to ask for help or clarification if needed. Keep up the good work!