Sprinters during a 200 m race

Sprinting involves a quick acceleration phase followed by a velocity maintenance phase. During the initial stage of sprinting, the runners have their upper body tilted forward in order to direct ground reaction forces more horizontally. As they reach their maximum velocity, the torso straightens out into an upright position. The goal of sprinting is to reach and maintain high top speeds to cover a set distance in the shortest possible time. A lot of research has been invested in quantifying the biological factors and mathematics that govern sprinting. In order to achieve these high velocities, it has been found that sprinters have to apply a large amount of force onto the ground to achieve the desired acceleration, rather than taking more rapid steps.

Quantifying sprinting mechanics and governing equations

Human legs during walking have been mechanically simplified in previous studies to a set of inverted pendulums, while distance running (characterized as a bouncing gait) has modeled the legs as springs. Until recently, it had been long believed that faster sprinting speeds are promoted solely by physiological features that increase stride length and frequency; while these factors do contribute to sprinting velocities, it has also been found that the runner's ability to produce ground forces is also very important.

Weyand et al. (2000) ^[1] came up with the following equation for determining sprint velocity:

${\displaystyle V=f_{\text{step}}F_{\text{avg}}/W_{\text{b}}L_{\text{c}},}$

where ${\displaystyle V}$ is the sprint velocity (m/s), ${\displaystyle f_{\text{step}}}$the step frequency (1/s), ${\displaystyle F_{\text{avg}}}$ the average force applied to the ground (N), ${\displaystyle W_{\text{b}}}$ the body weight (N), and ${\displaystyle L_{\text{c}}}$ the contact length (m).

In short, sprint velocity is reliant on three main factors: step frequency (how many steps you can take per second), average vertical force applied to the ground, and contact length (distance your center of mass translates over the course of one contact period). The formula was tested by having subjects run on a force treadmill (which is a treadmill that contains a force plate to measure ground reaction forces (GRF)). Figure 1 shows approximately what the force plate readout looks like for the duration of three steps. While this equation has proved to be fairly accurate, the study was limited in the sense that data was collected by a force plate that only measured vertical GRF rather than horizontal GRF. This led some people to the false pretense that simply exerting a greater vertical (perpendicular) force to the ground would lead to greater acceleration, which is far from correct (See Morin studies below).

In 2005, Hunter et al.^[2] conducted a study that determined relationships between sprint velocity and relative impulses in which gait and ground reaction force data was collected and analyzed. It was found that during accelerated runs, a typical support phase is characterized by a breaking phase followed by a propulsive phase (-FH followed by + FH). A common trend in the fastest subjects tested was that there was only a moderate to low amount of vertical force and a large amount of horizontal forces produced. Post study, it was hypothesized by the author that braking forces are necessary to store elastic energy in muscle and tendon tissue. This study loosely confirmed the importance of horizontal as well as vertical GRF during the acceleration phase of sprinting. Unfortunately, since data were collected at the 16-m mark, it was insufficient to draw definite conclusions regarding the entire acceleration phase.

Morin et al. (2011) ^[3] performed a study to investigate the importance of ground reaction forces by having sprinters run on a force treadmill that measured both horizontal and vertical ground reaction forces. Belt velocity was measured for each step and calculations were performed to find the “index of force application technique”, which determines how well subjects are able to apply force in the horizontal direction.

The second half of the test involved subjects performing a 100-m sprint on a man-made track using radar to measure the forward speed of runners to create velocity-time curves. The main result of this study showed that the force application technique (rather than simply the total amount of force applied) is the key determinant factor in predicting a sprinter's velocity. This has yet to be integrated into the governing equation of sprinting.

Kinetics

The kinetics of running describes the motion of a runner using the effects of forces acting on or out of the body. The majority of contributing factors to internal forces comes from leg muscle activation and arm swing.

Energetics

Di Prampero et al.^[9] mathematically quantifies the cost of the acceleration phase (first 30 m) sprint running through experimental testing. The subjects sprinted repeatedly on a track while radar determined their velocity. Additionally, it has been found in previous literature ^[10] that the energetics of sprinting on flat terrain is analogous to uphill running at a constant speed. The mathematical derivation process is loosely followed below:

In the initial phase of sprint running, the total acceleration acting on the body (${\displaystyle g'}$) is the vectoral sum of the forward acceleration and earth's acceleration due to gravity:

${\displaystyle g'=((a_{\text{f}}^{2}+g^{2}))^{0.5}}$

The “Equivalent slope” (ES) when sprinting on flat ground is:

${\displaystyle {\text{ES}}=\tan \left(90-\arctan \left(g+a_{\text{f}}\right)\right)}$

The “Equivalent normalized body mass” (EM) is then found to be:

${\displaystyle {\text{EM}}={\frac {g'}{g}}=\left[\left({\frac {a_{\text{f}}^{2}}{g^{2}}}+1\right)\right]^{0.5}}$

Following the data collection, the cost of sprinting (${\displaystyle C_{\text{sr}}}$) was found to be:

${\displaystyle C_{\text{sr}}=((155.4\,{\text{ES}})^{5}-(30.4\,{\text{ES}})^{4}-(43.3\,{\text{ES}})^{3}+(46.3\,{\text{ES}})^{2}+19.5\,{\text{ES}}+3.6)\,{\text{EM}}}$

The above equation does not take wind resistance into account, so considering the cost of running against wind resistance (${\displaystyle C_{\text{aer}}}$), which is known to be:

${\displaystyle C_{\text{aer}}=k'v^{2}}$

We combine the two equations to arrive at:

${\displaystyle C_{\text{sr}}=((155.4\,{\text{ES}})^{5}-(30.4\,{\text{ES}})^{4}-(43.3\,{\text{ES}})^{3}+(46.3\,{\text{ES}})^{2}+19.5\,{\text{ES}}+3.6)\,{\text{EM}}+k'v^{2}}$

Where ${\displaystyle g'}$ is the acceleration of the runner's body, ${\displaystyle a_{\text{f}}}$ the forward acceleration, ${\displaystyle g}$ the acceleration of gravity, ${\displaystyle k'}$ a proportionality constant and ${\displaystyle v}$ the velocity.

Fatigue effects

Fatigue is a prominent factor in sprinting, and it is already widely known that it hinders maximal power output in muscles, but it also affects the acceleration of runners in the ways listed below.

References