Running at altitude: the 100-m dash

Abstract

Purpose: Theoretical 100-m performance times (t_100-m) of a top athlete at Mexico-City (2250 m a.s.l.), Alto-Irpavi (Bolivia) (3340 m a.s.l.) and in a science-fiction scenario "in vacuo" were estimated assuming that at the onset of the run: (i) the velocity (v) increases exponentially with time; hence (ii) the forward acceleration (a_f) decreases linearly with v, iii) its time constant (τ) being the ratio between v_max (for a_f = 0) and a_{f max} (for v = 0).

Methods: The overall forward force per unit of mass (F_tot), sum of a_f and of the air resistance (F_a = k v², where k = 0.0037 J·s²·kg^-1·m^-3), was estimated from the relationship between a_f and v during Usain Bolt's extant world record. Assuming that F_tot is unchanged since the decrease of k at altitude is known, the relationships between a_f and v were obtained subtracting the appropriate F_a values from F_tot, thus allowing us to estimate in the three conditions considered v_max, τ, and t_100-m. These were also obtained from the relationship between mechanical power and speed, assuming an unchanged mechanical power at the end of the run (when a_f ≈ 0), regardless of altitude.

Results: The resulting t_100-m amounted to 9.515, 9.474, and 9.114 s, and to 9.474, 9.410, and 8.981 s, respectively, as compared to 9.612 s at sea level.

Conclusions: Neglecting science-fiction scenarios, t_100-m of a world-class athlete can be expected to undergo a reduction of 1.01 to 1.44% at Mexico-City and of 1.44 to 2.10%, at Alto-Irpavi.

Keywords: Acceleration; Air resistance; Altitude; Mechanical power; Sprint.