First, let's consider the prior art, the Ram Accelerator.
These are a type of mass accelerator which consist, more
or less, of a long tube which is filled with a mixture of fuel
and oxidizer, with thin, disposable diaphragms at each end
to keep the fuel/air mix contained.
A specially shaped
projectile is shot through one of the
diaphragms, the gas inside the tube is compressed between
the projectile and the tube walls, the compression causes
heating, and the heating causes gas to combust. If the
projectile is moving faster than the flame propagation
speed, most of the combustion occurs immediately behind
the projectile.
Since the gas pressure behind the projectile is higher than
in front, it continuously accelerates for as long as it's in the
tube.
This idea is to choose non electrically conductive materials
for the diaphragms, a conductive material for the tube,
and string a conductive wire from one diaphragm to the
other.
Next, we electrify the wire and tube, supplying a high DC
voltage which is slightly less than the breakdown voltage
of the fuel/air mix. We make sure that our power supply
can supply a nice, high current, once an arc is established.
Finally, we shoot a hollow conductive projectile through
the tube, with a high starting velocity. As it pierces the
first diaphragm, inertia keeps the wire in place, as the
projectile is going to fast to give the wire a chance to fall.
Since the tube is filled with combustible gas, and the
projectile is appropriately shaped, the same thing begins
to happen as occurs in a ram accelerator.
Since the starting voltage is so high, and the projectile is
conductive, electricity arcs from the wire, through the
projectile, to the wall of the tube.
The electric arc adds thermal energy to the burning
fuel/air mix, and thus increases thrust.
In addition, there's a flow of electricity down the wire,
through the projectile, and back up the tube, which
creates a magnetic field. There is a current flow through
the projectile at 90 degrees through this field. Thus, we
have electromagnetic thrust.
How much speed can be achieved by adding together
combustion and high current, I do not know, but it would
be exciting to find out.