The shape of a cable strung within a span and hanging under the influence of its own weight is known as a Catenary. Attaching a load at any point on the cable alters its shape. Additionally, the cable will elastically stretch due to the fact that it is under tension. If the cable is infinitely resistant to stretching, so called, "Rigid", it will not stretch.
Background information can be found in topic "Rope Sag - Catenary" @ Link:
https://igkt.net/sm/index.php?topic=6887.0If the catenary for the zipline rider (the "Load") is calculated at each point along the zipline cable, the trajectory, that is, the path that the rider takes, for the zipline can be determined.
Attachment #1 is a graph showing an example of a zipline trajectory for a 100 kg load and two of the catenaries. This 30-meter zipline starts at a height of 3 meters and finishes at a height of 2 meters for a drop of 1 meter. The lowest point in the trajectory has a height of about half a meter. Given that the height is known at each point in the trajectory, the change in height from any starting point determines the potential energy, kinetic energy and speed at every point.
In attachment #2, the trajectories for a Rigid cable and Elastic cable for a 100 kg load are shown. The Rigid (Analytical) curve is for a cable that does not stretch and has a constant total length of 30.2 meters. This curve is elliptical, see page 33 of Link:
http://www.ropelab.com.au/files/physics.pdf (suggested to me by agent_smith) or even ABoK #2592. The Rigid (Numerical) data are calculations using an extremely large cable strand elastic modulus (essentially no stretch). The Elastic (Numerical) data are using a cable strand elastic modulus of 47,500 (newtons/millimeters squared). The Elastic cable has approximately 0.30 meters more sag than the Rigid cable.
Edit: The Rigid cable trajectories will not look exactly symmetrical or elliptical. This is due to the difference in the scaling of the Height and X-Position axes in the graphs.