Spaceship Slingshot Mechanics
by MA Lloyd (malloy00@mik.uky.edu)
[This was part of a longer thread on rec.games.frp.gurps, and I didn't resist to save that little gem.  Incanus]
delphihome@geocities.com originally asked:
Just out of curiosity, how the frag does the slingshot effect work? Doesn't conservation of energy apply to momentum, also?
To which MA Lloyd replied:
Yes. Unfortunately the term slingshot is thrown around for three very different sorts of maneuvers:

Interstellar starship around a star. This is a simple turn using the gravity of the star in the same way as you can change the direction of your motion by grabbing a post and swinging around it. The ship leaves with exactly the same speed it arrived, but in a different direction. It's pure gravitation, you are in free fall the entire time. The maximum angle you can turn through is smaller the faster you are moving, larger the closer you can get to the star.

Gravity well burn. Essentially the same as above, but at the bottom of the gravity well you fire an engine. The advantage of doing this at the bottom of the gravity well, closest to the star/planet/whatever is you recover some of the potential energy of the burned fuel as kinetic energy. Call V_{i} your speed in interstellar space, V_{e} the escape velocity of the star at the point of closest approach and dV the deltav you can get from the engine. In free space you fire the engine and end up with speed V_{i} + dV. If instead you fall in toward the star, at periastron you have velocity equal to sqrt (V_{i}^{2} + V_{e}^{2}). Burn the engine and your speed is V_{p} = (V_{i}^{2} + V_{e}^{2})^{0.5} + dV, after escaping again from the star your speed is sqrt (V_{p}^{2}V_{e}^{2}), which is equal to sqrt {(V_{i}+dV)^{2} + 2V_{i}*dV*[(1+(V_{e}/V_{i})^{2})^{0.5} 1]}, which is larger than V_{i}+dV (rewrite it as sqrt[(V_{i}+dV)^{2}]) because that second term under the radical is always positive. Solar gravity does absolutely nothing to compensate.
 Three body maneuver. This one depends on having two gravity sources orbiting each other, and is the normal one used by space probes for example, where the three bodies are the Sun, the planet and the probe. You steal momentum of the planet about the sun to increase your speed relative to the sun without changing it relative to the planet. It's probably clearest with an analogy. Consider an experimenter with an ideal superball standing next to a railroad track. He throws the ball at a stationary train and it bounces back to him with the same speed. Now the train starts to pull away, the speed of the ball when it hits the train is smaller relative to the train, it still bounces back with the same speed it had relative to the train, but that's slower relative to the experimenter. At the limit he throws it at the same speed as the train, it hits with zero velocity, bounces back with zero velocity and falls to the base of the wall, remaining with the train. For the usual case of a probe, you want a velocity boost, so picture the train coming toward the experimenter, the ball bounces off the oncoming train and hits the experimenter harder than he threw it. If you think about it it's clear the limitation of this method is you can't add more speed to the ball than the speed of the train (speed of the planet relative to the sun) and you only get to add velocity in the direction of motion of the planet. Of course you can do a burn at pericenter here to and pick up a velocity boost in the same way as (2).