SpaceportTorque
Roger Arnold's Spaceport captures incoming vehicles at perigee and accelerates them (while it decelerates). The vehicles arrive at a fraction of Spaceport orbital velocity and speed up to orbital velocity as the Spaceport slows down. The spaceport is moving slightly faster than circular orbital perigee in the beginning, and thus has a small net-upward acceleration, which we will ignore for now.
For ease of calculation, we will assume the spaceport has a perigee velocity v_p = 8000 m/s, the vehicle arrives at 4000 m/s, and gee at altitude is a = 10 m/s² (it is 8.9 m/s² at 310 km equatorial altitude, but let's keep this simple). The vehicle acceleration a_v within the spaceport is assumed to be 5 gees or 50 m/s², or 5 a. The spaceport is assumed symmetrical around its center of mass, extending a distance -L towards the inlet and +L towards the terminus. The vehicle mass is M and the spaceport mass is 2 m L >> M , with mass evenly distributed along its length.
The stopping distance is 2 L = {v_p/2}^2 / ( 2 a_v ) so L = {v_p}^2 / ( 16 a_v ) = {v_p}^2 / ( 80 a ) = 80 km. The total length and stopping distance is 2 L or 160 km. We will arbitrarily assign the datum at the center, so the entry point at t = 0 is x = -L . The velocity of the vehicle is v = 4000 + 50 t and the acceleration to 8000 m/s takes 80 seconds. The distance is the integral of this velocity added to -80000m, so d = -80000 + 4000 t + 25 t^2 .
The vehicle "lift acceleration" relative to the Spaceport at perigee is a_l = ( v^2 - {v_p}^2 / r_p where r_p is the Liftport perigee radius and v_p is the perigee velocity. A liftport begins vehicle capture at the perigee of a slightly elliptical orbit; the momentum transferred to the vehicle reduces Liftport apogee.
If vehicles descending from the Moon are the main source of makeup momentum, they may arrive weeks apart, as the Moon is in a slow inclined orbit that crosses the liftport orbital plane twice a month. Hence, the Liftport must store a lot of momentum, for weeks. Either the Liftport apogee (and perigee velocity) is somewhat high, or it must be massive, or capture rates are limited, or it must burn a lot of rocket propellant; all these options are inconvenient.
So; a_l is significant, and negative, until the vehicle accelerates to Liftport velocity v_p at the far end. The vehicle is