The problem is not very well stated. The equilibrium compression
of the spring can be obtained by setting the restoring force equal to the
gravitational force, i.e. kx = mg. If the person transfers his weight
to the spring very slowly and gently, this also is approximately the
maximum the maximum compression.
If he transfers his weight suddenly to the spring, even though his
initial vertical velocity is zero, then the maximum compression is
determined by setting his change in gravitational potential energy mgx =
0.5kx^2. However, the net force at this maximum compression will be
upward, so he will oscillate. Eventually, the friction in the spring will
damp out the oscillations and he will end up at the equilibrium point....
Mark
Dr. Mark H. Shapiro
Physics Department
California State University, Fullerton
P.O. Box 6866
Fullerton, California 92834-6866