Location of minimum change in velocity for orbit escape

[majorbromly]

Homework Statement

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

Homework Equations

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

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.

 

[cepheid]

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).

 

[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.

 

[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.

 

[gneill]

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?

 

[majorbromly]

Ahh, I think I have it. Thanks!