
The
basic concepts of STL travel are covered by Newtonian physics. A
constant acceleration gives x(t)=½at^{2}+v_{0}t+x_{0},
where x is position, v velocity, a acceleration, and t time. This holds
very well for low speeds and objects that don't move near light speed,
don't shoot much propellant out the back, and don't have significant
drag forces to deal with.
In general, there are two cases we like to consider  leasttime
intercept, and zerozero intercept. A leasttime intercept means full
acceleration in one direction or another, getting to the target in as
little time as possible. There's one small problem with using leasttime
intercepts  you'll go right by the target, or through it. A zerozero
intercept calculates for starting and ending at v=0, and when reversed
to calculate acceleration from time and displacement, gives twice the
acceleration of a leasttime intercept. If there's significant relative
velocity between the two objects initially, neither model is
appropriate, and one must at least solve the quadratic with the
appropriate v_{0} term. If acceleration is variable, or the
vessel experiences some drag force, a more powerful differential
approach is required.
When the factors approach lightspeed, Newton's model of physics no
longer holds. Instead, we must work with Einstein's equations, p=γmv and E=γmc^{2}, where γ=(1v^{2}/c^{2})^{½}.
As γ~1 when v=0, the resting energy (mass energy) is E=mc^{2}, while the remainder
 T=(γ1)mc^{2}  is
kinetic. For massless particles, E=pc.
In either case, momentum and total energy are conserved.
There are, then, obstacles in space drives. First, any drive is limited
with respect to its energy vs momentum curve. To produce momentum, you
need energy, and mass. Throwing mass out the back only gives a high
quantity of momentum at low speeds, which requires lots of propellant.
Throwing it out fast requires lots of energy  up to 300 megajoules per
1 kgm/s of momentum on the high end case.
A pure fusion rocket can theoretically develop nearly 40 million kgm/s
per kg of burned fuel of thrust. This is not a significant limitation
for low speeds and craft carrying adequate fuel supplies; burning 1% of
your mass in hydrogen fuel and ejecting it as iron at 100% energy
efficiency will get you a Δv of ~40,000 m/s; burned over a hundred
seconds, that's still ~400g.

