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Recent models of star formation suggest that the mass of a star is largely determined by its history of motion through its natal molecular cloud. Such motions may be observable in the early stages of star formation. We have looked for the relative shifts of line-center velocity in low [(CO)-C-13 (1-0) and (CO)-O-18 (1-0)] and high [N2H+ (1-0)] density tracers toward a sample of 42 low-mass star-forming cores. Our results indicate that any motions of the high-density cores with respect to their low-density envelopes are very small (less than or similar to0.1 km s(-1)) compared to the motions expected from models of ballistic movement. We therefore conclude that isolated cores do not generally move ballistically with respect to their surrounding envelopes.