Location of minimum change in velocity for orbit escape

1. Dec 6, 2011

majorbromly

1. The problem statement, all variables and given/known data
A rocket is in elliptic orbit about the earth. To put it into escape
velocity, its engine is fired briefly, changing the rocket’s velocity by
deltaV . Where in the orbit, and in what direction, should the firing occur
to attain escape with a minimum value of deltav

2. Relevant equations
E=-c/A
Vescape=sqrt(2GM/r)

3. The attempt at a solution
I honestly have not been able to get anything meaningful.
I attempted to set kinetic and potential energies equal, as that's when the orbit becomes parabolic and escapes, but I have not really been able to glean any information about where the rockets should be fired.

It's quite obvious to me that it should be one of the two end points. But the relative interplay between K and U is something I can't quite prove.

Any help would be appreciated.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Dec 6, 2011

cepheid

Staff Emeritus
Your equation for escape velocity says that the farther away you are, the smaller your escape velocity is. I imagine that the best point would be at apogee (the farther point of the orbit).

3. Dec 6, 2011

majorbromly

That makes sense. However, at the closest point in the orbit, the object will be going much faster as well...which may or may not compensate for the increase in escape velocity. My problem is essentially that.

4. Dec 7, 2011

majorbromly

I have a decent proof for the direction of the thrust (although this is obvious in the first place of course). But I just cannot mathematically prove if it should be the apogee or perigee. It seems that either is equally as valid.

However, I've seen elsewhere that the rocket should be fired at it's closest pass to the planet.

5. Dec 7, 2011

Staff: Mentor

Calculate the orbit speed at each of your candidate points. Also calculate the escape velocity at that radius. What are the required Δv's at each point?

As for the direction, for a Δv that occurs over a small time interval (i.e. essentially an impulse), how would you combine the initial velocity and the Δv to find the final velocity vector? What Δv direction maximizes the sum?

6. Dec 7, 2011

majorbromly

Ahh, I think I have it. Thanks!