you realize Im sure that the acceleration does not have to be instantly at 5 m/s/s , that the acceleration could take place gradually to avoid collisions , however I assumed you would take that up.

lets drop all the confussing stuff.

we now have 1 mass and thats all.
we dont need a opposing rail gun.
we dont need to control the mass at all.
we just use the instant 5 m/s/s acceleration.

the mass is 100 kg
the pipes mass is 500 kg
the force supplied to the mass is 500kg-m/s/s or 500N
the pipes initial velocity is 0 m/s
the mass initial velocity is 0 m/s

we begin the mass acceleration
its initial velocity is 0 m/s
we accelerate it at 5m/s/s for a distance of 154.2 meters

accelerating this 1 mass results in a force of +3903.85N

to the pipe in the opposite direction.

the mass enters the first turnaround at a
velocity of 39.0385 m/s

it goes through the first turnaround
applies a force of -3903.85 N to the pipe.

it then free floats at a velocity of 39.0385 m/s
to the second turnaround.

it then goes through the second turnaround
and applies a force of +3903.85 N to the pipe.

the forces that apply to the pipe add up as follows

+3903.85 N acceleration
-3903.85 N 1st U turn
+3903.85 N 2nd U turn
+3903.85 N

its that simple.
the pipes initial velocity was 0 m/s
the pipe has a final velocity of 7.8077 m/s

the process repeats...

the pipes velocity increases every time the mass completes a cycle.

3/4 inch of dust build up on the moon in 4.527 billion years,LOL and QM is fantasy science.