And (because of pure physics) the planet that gave the sling-shot boost to the space craft's velocity (wait for it) slows down. The planet's orbit also changes. Admittedly, these effects are extremely small because of the relative mass involved.
I would have thought that as far as the orbital mechanics are concerned the planets used for slingshotting are sufficiently massive that this could be considered a sequential 2-body problem (and hence solvable) rather than a true N-body problem. I was never really a rocket scientist though, so please feel free to correct me if I'm wrong.
This can still be solved exactly (though now in the frame of the center of mass of the slingshott-er and slingshot-ee, which differs ever so slightly from the frame of the massive slingshott-er). It's when you start considering interactions with other bodies like the Sun and Jupiter that things start to get really messy.
One thing that will help is that it's easier to escape Mars than it is to escape Earth.
Nah, we need to learn how now so that when we need to go we can. Wouldn't it be great if we had colonies on Mars and a huge meteor destroyed all like on Earth. Elon would be known forever, or until a meteor hit Mars.
Explore is fine.
Not too worried about natives on Mars though I suppose it's possible.
No need to stop the acceleration. Simply use the thrusters to turn the ship 180 degrees while the main engines continue to thrust. In the few seconds of reversal process, the ship would only stray slightly from it's required path which could be easily corrected on the remainder of the trip. The captain might even choose not to inform the passengers of the maneuver. If they weren't looking out at the stars, passengers might not notice the slight acceleration of the turning thrusters. (Heh, heh, who was it who said "Do it at night?" - that way the passengers would be asleep.)
