During the first phase of the jump, before the cord is fully extended, I treat the jumper as if they are in free fall, including the effect of gravity and air resistance, but ignoring the interaction between the jumper and the cord.
It turns out that this interaction is non-negligible. As the cord drops from its folded initial condition to its extended final condition, it loses potential energy. Where does that energy go? It is transferred to the jumper!
The following diagram shows the scenario, courtesy of this web page on the topic:
The acceleration of the jumper turns out to be
where a is the net acceleration of the jumper, g is acceleration due to gravity, v is the velocity of the jumper, y is the position of the jumper relative to the starting point, L is the length of the cord, and μ is the mass ratio of the cord and jumper.
For a bungee jumper with mass 75 kg, I've computed the trajectory of a jumper with and without the effect of the cord. The difference is more than two meters, which could be the difference between a successful jump and a bad day.
The details are in this Jupyter notebook.
