Walking approximately 5.6 km/h (3.5 mph) is the most efficient speed. Running efficiency appears to taper off at higher speeds.
Transitioning from walking to running requires an increase of energy unlike once speculated. The primary stimulus for the transition from walking to running [Preferred Transition Speed (PTS)] is prompted by the perceived or imminent fatigue and discomfort in the tibialis anterior and other dorsal flexion muscle of the ankle (Hreljac 1995, Prilutshy et al 2001).
Usain Bolt was the fastest human on record during the 100 meter sprint at 44.72 km/h (27.79 mph). That was the average speed between the 60th and the 80th meter.
Calories calculated from regression formulas adapted by ACSM (ACSM's Guidelines for Exercise Testing and Prescription). See Walk/Run Metabolic Calculator based on these formulas.
Hreljac A (1995). Determinants of the gait transition speed during human locomotion. Journal of Biomechanics. 28: 669-677.
Prilutsky BI, Gregor RJ (2001). Swing-and support-related muscle actions differentially trigger human walk-run run-walk transitions. Journal of Experimental Biology. 204: 2277-2287.