Proving the slingshot effect causes less speed for a space shuttle

[Tommy1995]

When a rocket approaches a planet from the front of it's orbit such as in the bottom drawing shown in the link <http://img156.imageshack.us/img156/724/slingshotph4.jpg>, the gravitationally assisted manoeuvre or slingshot affect will cause the shuttle's final velocity to be less than the initial velocity, which makes logical sense. However, I can't however seem to prove through using the law of the conservation of momentum that the final velocity of the shuttle will be less than the initial velocity.

Please help prove this concept through the law of the conservation of momentum.

 

[Danger]

Hi, Tommy;
Welcome to PF. The Shuttle, as in the Space Transportation System, never had an opportunity to utilize the slingshot effect. Although I'm not sure, I don't think that even the Mars probes did so. Voyageurs 1 and 2 did, but around much larger planets than we have in our immediate neighbourhood.
The principle is that an object will accelerate in a gravitational well, and might escape by skipping off of that well and extracting a bit of juice on the way.
It's sort of like the "Supercharger" that I had for my Hot Wheels track when I was a kid. A couple of counter-rotating foam rubber tires that you fed the cars into... they went in at about 20 kph (hand-fed) and exited at about 60 kph (scars in the wall to prove it; they didn't corner well).
Anyhow, the point is that the acceleration due to gravity can be used to advantage as long as the package in question has been directed properly. It gains speed on the way down, and the planet gives up an unnoticeable percentage of its angular momentum to the object's velocity (and keep in mind that "velocity" is a vector).

edit: I haven't looked at your illustration, because I had trouble accessing it. I'll try again later, and might alter my original opinion according to what I see there.

 

[mfb]

If you approach a planet from its back (as seen in the reference frame of the sun), make a turn around it and go back again, the planet accelerated a tiny bit (due to momentum conservation). In the frame of the planet, the speed of the rocket is the same afterwards - but in the frame of the sun, you changed the direction from "in the same direction as the planet" to "in the opposite direction".

If you approach a planet from its front, it is just the other way round, and the rocket gains speed.

 

[Tommy1995]

Danger, the information that I've read about this concept says that the effect of gravity accelerating the err spaceship :P, has very little to do with the increased velocity from the slingshot effect. Rather it states that its all about the orbital velocity of the planet and the vector addition between the velocity of the spaceship relative to the planet and the orbital velocity of the planet.