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Fig. 1. A model of how stopping ability and total travel distance could constrain modulation of velocity during a start and stop. The hypothetical movements in A and B have identical starting accelerations (a_start=slope=1) and total displacements (S_total=areas under the triangles or trapezoids with a vertex indicated by a circle=3 units), but the stopping acceleration (a_stop) varies among the cases indicated by the different colors within each panel. (A) Starting acceleration continues up to the instant when stopping begins (indicated by circles). Compared to a_stop=1, increased stopping ability (green) allows more distance to accelerate to a greater maximal velocity (V_max), which decreases total travel time. Decreased stopping ability (red) has detrimental effects on V_max and total travel time. Consequently, the average velocities (S_total divided by total time) for a_stop=2 and 0.5 are 115% and 81% of the value when a_stop=1, respectively. (B) When the total distance provides sufficient time so that a physiologically maximum speed is attained and momentarily sustained, stopping ability will not affect V_max. However, increased stopping ability decreases total time and hence the average velocities for a_stop=2 and 0.5 are 107% and 89% of the value when a_stop=1, respectively. Maintaining a constant velocity in between the starting and stopping accelerations (B) increases total travel time and hence decreases average velocity compared to beginning a stop immediately after the cessation of a starting acceleration (compare A vs B for equal values of a_start and a_stop). If the only objective of starting and stopping is to minimize total travel time (and maximize average speed) for a given distance, then maximal accelerating and decelerating capacities should be used. (C) For a linear increase in velocity followed immediately by a linear decrease in velocity as in A, V_max=a_start[(2S_total)/(a_start+1/a_stop)]^0.5 and hence the upper limit of V_max is a_start^0.5(2S_total)^0.5. Axes show arbitrary units.

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