How Does the Vis-Viva Equation Explain Orbital Speed Changes to Reach the Moon?

[Lucy Yeats]

Homework Statement

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)

Homework Equations

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

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?

 

[D H]

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.

 

[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!

 

[Lucy Yeats]

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

 

[asteriatic]

It appears that you have made a computational error in your attempt at a solution. The correct equation for the speed at perigee (vp) is vp^2 = GM((2/r0)-(1/ra)), as you have stated. However, the equation for the speed at apogee (va) is va^2 = GM((2/r0)-(1/ra)), not v0^2 = GM((2/r0)-(1/r0)). This is because the speed at apogee is not equal to the speed at the initial circular orbit, but rather is the speed at the point where the spacecraft is farthest from the Earth.

Using the correct equation for va, we get:

vp^2/v0^2 = (GM((2/r0)-(1/ra)))/(GM((2/r0)-(1/r0))) = ((2/r0)-(1/ra))/((2/r0)-(1/r0)) = (2ra-r0)/(2r0-r0) = (2ra-r0)/r0

This can be simplified to (2ra)/(2r0) = ra/r0

Substituting this into the original equation, we get:

(vp/v0)^2 = 2(ra/r0) / (1 + (ra/r0)) = 2ra/(r0+ra)

This matches the desired result, so it appears that the error was in the equation for va.