How much time would one have to attach to the tether's end? Since it is connecting two different orbits then I'm imagining that it would be fairly brief. If one misses the connection, then what?
I liken a catch at apoapsis to catching a ball at the top of it's bounce. For a brief time, the ball hangs motionless -- and then gravity pulls it back down. The less the acceleration, the longer the ball will hover at the top of a toss.
Regions of the tether that feel a substantial net acceleration will have a greater need for fast reflexes and good timing. The regions of the tether closer to the balance point can catch at a more relaxed space. Catching at the balance point would be like docking with the I.S.S.
For an example I will use the ellipse common to the Phobos and Deimos tether:
Note: When I use directional words like top, foot, above, below, up or down, I'm using Mars as the center. Down means Marsward.
Making the catch at the Deimos tether foot
Both the payload and Deimos tether foot are traveling about 1.18 km/s. But it is the relative velocity that counts. After all, I am traveling 30 km/s as earth circles the sun and so is my computer monitor. Do I worry about a catastrophic collision with my computer monitor? Not since I'm moving about zero km/s with regard to my computer.
Catching at the foot of the Deimos tether:
1 minute before the catch, the relative speed is only about 5 mph.
I'll compare this to driving a car. Driving down the road at a leisurely 30 mph, I step on the brakes. I don't stomp on them, mind you. Just decelerating at my usual careful pace, it takes 6 seconds to come to a full stop. Now look at the payload 5 minutes before the catch: 22.6 mph. Compared to my Sunday driving, this payload is moving like a turtle coming out of hibernation.
From Phobos throw to Deimos catch is an ~8 hour trip. During the vehicle can make measurements of it's distance and velocity with regard to the Deimos tether foot and compare it to optimal distance and velocity.
If catching below a tether's balance point, the payload would rendezvous with the tether at the trailing edge. If the tether is in a prograde orbit, the payload would land on the western end of a ramp:
In this cartoon I have a quadpod on wheels entering on the west end of the ramp. The wheels are only partially for comic relief. Wheels would actually be helpful landing on a platform.
The quadpod is a fanciful design not really relevant to tethers. I use it because it can quickly make slight adjustments to speed along any direction. The closing velocity is almost completely vertical. There will be a time when the space ship is quite close to the tether and falling up at a high speed. So it would be good to be able to do a slight tap on the brakes or gas pedal.
Also to give a little error room to the landing on the west ramp, I imagine a folding west end of the ramp that can extend itself after the ship has matched altitudes.
Acceleration, net weight
At this part of the Deimos tether, centrifugal acceleration is -.0681 m/s2 and Mars gravity is .1017 m/s2. Net acceleration is .034 m/22. A Sumo wrestler weighing 400 pounds on earth's surface would weigh 1.4 pounds. A coin falling out of your coat pocket would take ~8 seconds to reach your foot (assuming distance from coat pocket to foot is one meter).
Making the drop to Phobos
To send a payload on it's way to Phobos, simply roll of the east edge of the ramp.
Making the catch at the Phobos tether top
Catching at the Phobos tether:
1 minute out, the tether is moving 18 mph. A faster pace than the Deimos tether catch, but still much more leisurely than me rolling my car into the driveway from a Sunday drive.
Acceleration, net weight
At this part of the Phobos tether, centrifugal acceleration is -.536 m/s2 and Mars gravity is .4027 m/s2. Net acceleration is -.133 m/22. A Sumo wrestler weighing 400 pounds on earth's surface would weigh -5.44 pounds. A coin falling out of your coat pocket would take 4 seconds to reach your foot (assuming distance from coat pocket to foot is one meter).
From the point of view of someone on Mars, the acceleration would seem upward. Looking through a telescope, they'd see the Sumo wrestler bumping against the ceiling like a helium balloon.
Making the drop to Phobos
To send a payload on it's way to Deimos, simply roll of the west edge of the ramp.
Some general rules for vertical tethers
This is an illustration for any vertical tether in a circular orbit. It also applies to Clarke style beanstalks.
The red orbits below the circular balance point's orbit move faster than the tether except where they cross the tether. At crossing points the orbits move the same speed as the tether. I explain here how tether matching orbits are found.
For points on the tether below the balance point's circular orbit, entrance/catching ramps are on the trailing edge of the tether (the west end for tethers in prograde orbits). To drop to lower orbits, roll off the leading edge (east end of the ramp in prograde orbits).
For points on the tether above the balance point's circular orbit, entrance/catching ramp are on the leading edge of the tether. To throw to high orbits, roll off the trailing edge.
Making catches in steep acceleration gradients
Things are less relaxed as the tether extends further from the balance point. As soon as I have time, I will look at a Phobos tether whose foot extends into Mars upper atmosphere. The net acceleration at this foot would be about 3 km/s2 or about a third of an earth g.