Wiley E. Coyote Answer

The speed of sound in rubber is about 60 m/s. The fact that the
rope detaches from the bridge has to reach Wiley before anything
happens. Since he is 200 m down, this cannot happen for 0 < t < 2.
Hence w stays at w = -200m during this time interval.

If you find this hard to believe, check:
https://www.youtube.com/watch?v=8UimHnsWSBc
or
https://www.youtube.com/watch?v=JsytnJ_pSf8
or
https://vimeo.com/96786001

Note #1.
See the introduction to the problem "Experiments with a slinky"
(problem #4 in problem set #7) where the equation for an elastic
rod under gravitational tension is derived. This is the equation
for a slinky, and (roughly) the situation with Wiley E. Coyote.

Note #2.
In addition to the longitudinal waves, a bunjee cord (or a slinky)
can also propagate torsional and transversal waves [with different
speeds]. But none of them affect the tension in the cord, which is
what supports Wiley at 200 m below the bridge.

Note #3.
All of this is follows from elasticity (a continuum approximation).
In principle, at least, there could be (at the microscopic) other
signals/waves propagating faster than the elastic waves [I do not
know]. But they are tiny, and would not change the situation for
Wiley E. Coyote in any noticeable way.
EOF