[Harri (OP)]

Just a question I was asked by a friend that I didn't have an answer for! He asked, if space beyond our atmosphere is empty, just how does a vehicle alter it's speed or direction? Assuming a vehicle has some kind of propulsion and method of changing direction, what does it propel against?

[Halc]

Just a question I was asked by a friend that I didn't have an answer for! He asked, if space beyond our atmosphere is empty, just how does a vehicle alter it's speed or direction? Assuming a vehicle has some kind of propulsion and method of changing direction, what does it propel against?

It propels against the mass of its own fuel, simple as that.
The faster it can eject this reaction mass, the more thrust it gets from a given Kg of fuel.

[Harri (OP)]

It propels against the mass of it's own fuel! Brilliant!

[Janus]

It propels against the mass of it's own fuel! Brilliant!

It is simply the law of action and reaction.  In order to eject the fuel at high speed out of the nozzle, the rocket has to exert a force on it in that direction.  This, in turn, also results in an equal force pushing the rocket in the other direction.
There are a couple of misconceptions going around about rockets.  One is that they need something Air or something outside of the rocket to "Push against" in order work, which is not true.  If you were floating in the vacuum of space, holding a brick, and threw the brick away from yourself, you would move start moving in the opposite direction to the brick.  It's a matter of conservation of momentum.  Both you and the brick together start with zero momentum total.  You toss the brick to the right, giving it a non-zero momentum to the right.  In order for the combination of you and brick to maintain a total momentum of 0, you need to move at some speed to the left so your momentum to the left cancels out the brick's momentum to the right.
In reality, it is a disadvantage to have an atmosphere outside of the rocket. The outside air pushing in on the rocket nozzle decreases the speed at which the gasses can escape, and reduce the effective thrust of the rocket.

The other misconception is that a rocket's top speed is limited to the to its exhaust speed.   As long as a rocket is firing and expelling exhaust, it will continue to accelerate.  The only thing that limits its final velocity ( other than the Relativistic limit of the speed of light), is how much fuel it carries.

[Jaaanosik]

It propels against the mass of it's own fuel! Brilliant!

It is simply the law of action and reaction.  In order to eject the fuel at high speed out of the nozzle, the rocket has to exert a force on it in that direction.  This, in turn, also results in an equal force pushing the rocket in the other direction.
There are a couple of misconceptions going around about rockets.  One is that they need something Air or something outside of the rocket to "Push against" in order work, which is not true.  If you were floating in the vacuum of space, holding a brick, and threw the brick away from yourself, you would move start moving in the opposite direction to the brick.  It's a matter of conservation of momentum.  Both you and the brick together start with zero momentum total.  You toss the brick to the right, giving it a non-zero momentum to the right.  In order for the combination of you and brick to maintain a total momentum of 0, you need to move at some speed to the left so your momentum to the left cancels out the brick's momentum to the right.
In reality, it is a disadvantage to have an atmosphere outside of the rocket. The outside air pushing in on the rocket nozzle decreases the speed at which the gasses can escape, and reduce the effective thrust of the rocket.

The other misconception is that a rocket's top speed is limited to the to its exhaust speed.   As long as a rocket is firing and expelling exhaust, it will continue to accelerate.  The only thing that limits its final velocity ( other than the Relativistic limit of the speed of light), is how much fuel it carries.

Janus,
would it be possible to design an inertial propulsion system?
A spaceship that does not eject anything but can accelerate?
What do you think?
Jano

[Bored chemist]

would it be possible to design an inertial propulsion system?
A spaceship that does not eject anything but can accelerate?
What do you think?

No.
That would be a breach of the conservation of momentum.

[syhprum]

In the thirties the popular way to propel space ships was with the use of antigravity shields but their manufacture proved problematical

[RD]

It propels against the mass of it's own fuel! Brilliant!

The accelerated mass ...

 https ://youtu.be/Q8Sv5y6iHUM?t=120

this method is a bit too discrete for my liking: bombs are being dropped out the bottom end.

[Jaaanosik]

would it be possible to design an inertial propulsion system?
A spaceship that does not eject anything but can accelerate?
What do you think?

No.
That would be a breach of the conservation of momentum.

Are you sure?   
Jano

[evan_au]

would it be possible to design an inertial propulsion system?
A spaceship that does not eject anything but can accelerate?

The electrodynamic tether has been suggested as a way to remove obsolete satellites from orbit, by accelerating them as they pass through the Earth's magnetic field.
- This mechanism works on the angular momentum of Earth's core, coupled to the satellite by Earth's magnetic field.

It is also possible to use them for station-keeping, but I think this application would be far too risky - imagine every satellite in low Earth orbit dangling a line several kilometers long - this would greatly increase the chance of satellite colissions.
- It would be like an insect flying through the Waitomo glow-worm caves, filled with dangling threads that will snare and kill you...
See: https://en.wikipedia.org/wiki/Electrodynamic_tether