How Do You Calculate Velocities for a Hohmann Transfer Orbit?

[hb243]

Homework Statement

A Hohmann transfer orbit is a way of transferring a spacecraft between two planetary
orbits (which we shall assume are circular) by using one half of an elliptical orbit about
the Sun.
A spacecraft is initially moving around the Sun with the orbital speed V1 of
the first planet, at radius R1, and it is to be moved to a radius R2. The orbital speeds of the spacecraft at perihelion (point A) and aphelion (point B) in the
elliptical orbit are vA and vB respectively. Write down the conditions on vA and vB coming
from (i) the conservation of energy and (ii) the conservation of angular momentum, on
the assumption that the gravitational fields of the planets have a negligible effect on the
spacecraft compared to the gravitational field of the Sun.
Hence derive the required to accelerate the spacecraft into the transfer orbit.

Homework Equations

The Attempt at a Solution

i) 0.5(Va)^2 - GM/R1 = 0.5(Vb)^2 - GM/R2
ii) R1*Va=R2*Vb

I don't know where to go from here? Also, sorry if this is in the wrong forum.

 

[gneill]

The Attempt at a Solution

i) 0.5(Va)^2 - GM/R1 = 0.5(Vb)^2 - GM/R2
ii) R1*Va=R2*Vb

I don't know where to go from here? Also, sorry if this is in the wrong forum.

Okay, so you've got conservation of the total mechanical energy and conservation of angular momentum. With these two equations you can determine what the velocities va and vb are in terms of GM, R1 and R2. This will give you your departure and arrival conditions on the transfer ellipse.

In order to "get on" and "get off" the transfer ellipse at each end you'll have to make a delta-V change. The first will turn the spacecraft 's circular orbit into the elliptical orbit, while the second will change the elliptical orbit back to a circular orbit.