  Once weakly-interacting massive particles (WIMPs) are unambiguously detected
in direct-detection experiments, the challenge will be to determine what one
may infer from the data. Here, I examine the prospects for reconstructing the
local speed distribution of WIMPs in addition to WIMP particle-physics
properties (mass, cross sections) from next-generation cryogenic and
liquid-noble direct-detection experiments. I find that the common method of
fixing the form of the velocity distribution when estimating constraints on
WIMP mass and cross sections means losing out on the information on the speed
distribution contained in the data and may lead to biases in the inferred
values of the particle-physics parameters. I show that using a more general,
empirical form of the speed distribution can lead to good constraints on the
speed distribution. Moreover, one can use Bayesian model-selection criteria to
determine if a theoretically-inspired functional form for the speed
distribution (such as a Maxwell-Boltzmann distribution) fits better than an
empirical model. The shape of the degeneracy between WIMP mass and cross
sections and their offset from the true values of those parameters depends on
the hypothesis for the speed distribution, which has significant implications
for consistency checks between direct-detection and collider data. In addition,
I find that the uncertainties on theoretical parameters depends sensitively on
the upper end of the energy range used for WIMP searches. Better constraints on
the WIMP particle-physics parameters and speed distribution are obtained if the
WIMP search is extended to higher energy (~ 1 MeV).
