We describe a time-of-flight apparatus that uses core extraction to determine nascent product laboratory velocity distributions from which differential cross sections may be deduced. We emphasize the characterization of the instrument, the reaction conditions, and the calibration procedure. For this purpose, we have measured H-atom velocity distributions from HBr photolysis, as well as the H-2(v(')=4,J(')=1) velocity distribution arising from the H+HBr reaction under quasi-monoenergetic collision conditions at 1.9 eV. Collisional energy spread and reagent internal state distributions were determined from the rotational and translational temperatures of the HBr photolytic precursor and the D-2 diatomic reagent. The differential cross section for H+D-2-->HD(v(')=2,J(')=3)+D at 1.55+/-0.05 eV is presented and found to peak near 145 degrees+/-10 degrees with an approximate full width at half maximum (FWHM) of 40 degrees.