# Elliptical orbits question

1. Feb 14, 2012

### Lucy Yeats

1. The problem statement, all variables and given/known data
An eﬃcient way to reach the Moon is to ﬁrst put the spacecraft in a low circular Earth
orbit (radius r0, speed v0). The speed is then boosted to vp giving an elliptical orbit with
apogee at the Moon’s orbit, ra, and perigee at r0. Show that:

(vp/v0)^2=2ra/(r0+ra)

2. Relevant equations

http://en.wikipedia.org/wiki/Vis-viva_equation

3. The attempt at a solution

Using the Vis Viva equations, I found:
vp^2=GM((2/r0)-(1/ra))
v0^2=GM((2/r0)-(1/r0))=GM(1/r0)

so (vp/v0)^2=((2/r0)-(1/ra))/(1/r0)
Which simplifies to (2ra-r0)/ra, which isn't right.

Where did I go wrong?

2. Feb 14, 2012

### D H

Staff Emeritus
The vis viva equation is $v^2 = \mu\left(\frac 2 r - \frac 1 a\right)$, where $\mu=GM$ is the gravitational parameter, $r$ is the radial distance, and $a$ is the semi major axis of the orbit.

Your mistake was using the apogee distance in lieu of the semi major axis.

3. Feb 14, 2012

### Lucy Yeats

I thought in this case the apogee distance was the semi major axis? :-/

If not, how do I find the semi major axis?

Thanks for helping!

4. Feb 14, 2012

### Lucy Yeats

Sorry, I've got it now! Thanks! :-)