  The softness of elastic atomic collisions, defined as the average number of
collisions each atom undergoes until its energy decorrelates significantly, can
have a considerable effect on the decay dynamics of atomic coherence. In this
paper we combine two spectroscopic methods to measure these dynamics and obtain
the collisional softness of ultra-cold atoms in an optical trap: Ramsey
spectroscopy to measure the energy decorrelation rate and echo spectroscopy to
measure the collision rate. We obtain a value of 2.5 (3) for the collisional
softness, in good agreement with previously reported numerical molecular
dynamics simulations. This fundamental quantity was used to determine the
$s$-wave scattering lengths of different atoms but has not been directly
measured. We further show that the decay dynamics of the revival amplitudes in
the echo experiment has a transition in its functional decay. The transition
time is related to the softness of the collisions and provides yet another way
to approximate it. These conclusions are supported by Monte Carlo simulations
of the full echo dynamics. The methods presented here can allow measurements of
a generalized softness parameter for other two-level quantum systems with
discrete spectral fluctuations.
