# Determinants of Top Speed Sprinting: Minimum Requirements for Maximum Velocity

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{c}(m) during the ground contact time t

_{c}(s) [3,15]:

_{c}is determined by leg length L

_{0}(m) and the total excursion angle during contact θ

_{c}(deg):

_{c}≈ 60 deg for humans and other bipedal runners [12,13,14]. Based on Equation (2), it can be approximated that L

_{c}≈ L

_{0}(see Figure 6 in [5]). Equation (1) then shows that t

_{c}is directly related to L

_{0}and Speed:

_{a}(s) is the time interval from takeoff of one foot until touchdown of the contralateral foot. Prior research has demonstrated that t

_{a}is approximately 0.12 ± 0.02 s for runners across a broad range of top speeds (~6 to 11 m/s) [2,3,4,5]. Step time t

_{step}(s) is the time interval to complete one step, and is equal to the sum of t

_{c}and t

_{a}:

_{step}:

_{Zavg}/mg) can be determined if t

_{c}and t

_{a}are known:

^{2}) is gravitational acceleration [16,17]. Given that t

_{c}is generally inversely related to top speed, and t

_{a}remains relatively constant across a range of runners and top speeds, F

_{Zavg}/mg generally increases with top speed [2,3,4,17].

_{total}, which is the total thigh range of motion from peak extension through peak flexion (Figure 1b). The frequency f (Hz) can be expressed in terms of the period T (s) for one full cycle of the leg or equivalently in terms of t

_{step}:

_{total}(deg) and the total excursion angle during contact θ

_{c}(deg) can then be derived from Equation (9) using the conditions θ(t) = θ

_{c}/2 at t = T/2 − t

_{c}/2, and T = 1/f:

_{avg}(deg/s) has been shown to increase linearly with top speed [5] and can be determined by:

_{max}(deg/s

^{2}) during the swing phase can be derived from the second derivative of Equation (9). This variable has also been shown to increase linearly with top speed [9] and can be determined by:

_{0}= 0.85 m, 0.95 m, and 1.05 m (greater trochanter to ground in standing position). Data for 13.0 m/s were calculated to explore the requirements for achieving this level of performance, with the recognition that no human runner has yet attained this speed. The theoretical calculations required only two simplifying assumptions across all conditions, θ

_{c}= 60 deg and t

_{a}= 0.12 s.

## 3. Results

_{c}), step rate (SR), step length (SL), ratio of step length to leg length (SL/L

_{0}), ratio of stance-averaged vertical force to body weight (F

_{Zavg}/mg), total thigh range of motion from peak extension to peak flexion (θ

_{total}), thigh angular velocity averaged throughout stride cycle (ω

_{avg}), and modeled maximum thigh angular acceleration during the swing phase (α

_{max}).

## 4. Discussion

_{c}(Figure 2a), aligning with values from several experimental data sets [1,2,3,5]. Calculations of SR and SL (Figure 2b,c) increased with Speed, and values were generally in agreement with prior investigations into top-speed sprinting [2,3,4,5]. The relationship between Speed and F

_{Zavg}/mg (Figure 2e) was positive and linear across all top speeds and corresponded to similar increases in the experimental data of Weyand et al. [2,3] across a range of top speeds in a heterogenous pool of subjects. A positive and linear relationship was also demonstrated between Speed and the thigh angular variables, with theoretical values of θ

_{total}, ω

_{avg}, and α

_{max}(Figure 2f–h) aligning with recent experimental data on thigh angular motion at top speed [5,9].

_{0}had a noticeable effect on the calculated outcome variables. Across the three hypothetical athletes, increased leg length allowed a given Speed to be attained with: (A) longer t

_{c}; (B) decreased SR; (C) longer SL; (D) decreased SL/L

_{0}ratio; (E) reduced F

_{Zavg}/mg; (F) decreased θ

_{total}; (G) slower ω

_{avg}; and (H) decreased α

_{max}. Thus, from a purely theoretical standpoint, it is clear that longer legs may allow for fast speeds to be attained with reduced mechanical requirements (i.e., prolonged ground contact times, decreased vertical forces, reduced thigh angular velocities and accelerations). This may in part explain the record-breaking performances achieved by Usain Bolt, whose unusually tall stature for a male sprinter likely provided specific advantages over his shorter competitors [10]. Of course, inherent biological tradeoffs clearly exist that interplay to govern the optimal dimensions for sprinting speed. As it relates to thigh angular motion, since torque is the product of moment of inertia and angular acceleration, and the moment of inertia is proportional to the mass and length of the leg (mL

_{0}

^{2}), the hip torque required to generate a given magnitude of thigh angular acceleration will escalate with increases in leg length. Therefore, while longer legs may allow for higher speeds to be attained with decreased requirements for the variables analyzed here (Figure 2a–h), longer legs may also require increases in other physical parameters such as torque generating capacity.

_{c}= 60 deg and t

_{a}= 0.12 s. If runners exhibit large deviations from these simplifying assumptions, there is the possibility for experimental data to not align with the values in Table 1, Table 2 and Table 3 and Figure 2. For the limb excursion angles selected during ground contact, prior research has indicated that these angles are likely constrained by leg extensor muscle effective mechanical advantage [3,19]. However, deviations from the assumed value of θ

_{c}= 60 deg will affect the other calculated variables, with increased θ

_{c}allowing for longer t

_{c}and decreased F

_{Zavg}/mg to achieve a given speed, and vice versa. As it relates to aerial time, prior investigations have found that even for runners of different body dimensions and top speeds that t

_{a}= 0.12 ± 0.02 s [2,3,4,5], and thus employing a standard t

_{a}here for hypothetical runners with varying L

_{0}likely did not introduce major errors. However, differences in individual running styles may exist [20,21] that could result in t

_{a}outside the standard range presented here, subsequently affecting the other calculated variables.

_{0}and Speed, this framework may serve as a blueprint for coaches and athletes to specifically evaluate mechanics based on their body dimensions and performance goals.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Symbols and Abbreviations

θ(t) | thigh angle as a function of time (deg) |

θ_{c} | total thigh excursion during the ground contact phase (deg) |

θ_{total} | total thigh range of motion from peak extension to peak flexion (deg) |

ω_{avg} | thigh angular velocity averaged throughout stride cycle (deg/s) |

α_{max} | modeled maximum thigh angular acceleration during swing (deg/s^{2}) |

F_{Zavg}/mg | ratio of stance-averaged vertical force to body weight mg |

f | frequency of thigh angular motion (Hz) |

g | gravitational acceleration (9.8 m/s^{2}) |

L_{c} | contact length (m) |

L_{0} | leg length (m) |

m | body mass (kg) |

SL | step length (m) |

SL/L_{0} | ratio of step length to leg length |

Speed | runner’s forward speed |

SR | step rate (steps/s) |

T | time period of thigh angular motion (s) |

t | time (s) |

t_{a} | aerial time (s) |

t_{c} | ground contact time (s) |

t_{step} | step time (s) |

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**Figure 1.**Leg angular motion during the gait cycle. (

**a**) Simplified illustration of harmonic oscillatory thigh motion that assumes symmetrical peak thigh flexion and extension values. (

**b**) Thigh angular position θ(t) during the gait cycle. Thigh angular kinematics as a function of time are determined by the parameters of frequency (f = 1/T, where T is the period of the cycle) and θ

_{total}, which is the total thigh range of motion from peak extension through peak flexion. The total excursion angle during contact (θ

_{c}) occurs in the contact time (t

_{c}) between t = T/2 − t

_{c}/2 and t = T/2 + t

_{c}/2.

**Figure 2.**Calculated data across a range of top speeds for the three hypothetical athletes of different body dimensions. Illustrated lines based on values in Table 1, Table 2 and Table 3, arranged by athlete leg length (L

_{0}). (

**a**) ground contact time (t

_{c}), (

**b**) step rate (SR), (

**c**) step length (SL), (

**d**) ratio of step length to leg length (SL/L

_{0}), (

**e**) ratio of stance-averaged vertical force to body weight (F

_{Zavg}/mg), (

**f**) total thigh range of motion (θ

_{total}), (

**g**) average thigh angular velocity (ω

_{avg}), (

**h**) modeled maximum thigh angular acceleration (α

_{max}).

Speed (m/s) | t_{c}(s) | SR (steps/s) | SL (m) | SL/L_{0}(ratio) | F_{Zavg}/mg(ratio) | 𝜃_{total} (deg) | ω_{avg}(deg/s) | α_{max}(deg/s ^{2} × 10^{3}) |
---|---|---|---|---|---|---|---|---|

7.00 | 0.121 | 4.14 | 1.69 | 1.99 | 1.99 | 84.5 | 349.8 | 7.15 |

8.00 | 0.106 | 4.42 | 1.81 | 2.13 | 2.13 | 89.2 | 394.3 | 8.60 |

9.00 | 0.094 | 4.66 | 1.93 | 2.27 | 2.27 | 94.1 | 438.6 | 10.09 |

10.00 | 0.085 | 4.88 | 2.05 | 2.41 | 2.41 | 99.0 | 482.8 | 11.62 |

11.00 | 0.077 | 5.07 | 2.17 | 2.55 | 2.55 | 104.0 | 526.9 | 13.18 |

12.00 | 0.071 | 5.24 | 2.29 | 2.69 | 2.69 | 109.0 | 571.1 | 14.77 |

13.00 | 0.065 | 5.39 | 2.41 | 2.84 | 2.84 | 114.0 | 615.2 | 16.38 |

Speed (m/s) | t_{c}(s) | SR (steps/s) | SL (m) | SL/L_{0}(ratio) | F_{Zavg}/mg(ratio) | 𝜃_{total} (deg) | ω_{avg}(deg/s) | α_{max}(deg/s ^{2} × 10^{3}) |
---|---|---|---|---|---|---|---|---|

7.00 | 0.136 | 3.91 | 1.79 | 1.88 | 1.88 | 81.0 | 316.9 | 6.12 |

8.00 | 0.119 | 4.19 | 1.91 | 2.01 | 2.01 | 85.2 | 356.9 | 7.38 |

9.00 | 0.106 | 4.43 | 2.03 | 2.14 | 2.14 | 89.5 | 396.6 | 8.68 |

10.00 | 0.095 | 4.65 | 2.15 | 2.26 | 2.26 | 93.8 | 436.3 | 10.01 |

11.00 | 0.086 | 4.85 | 2.27 | 2.39 | 2.39 | 98.2 | 475.8 | 11.38 |

12.00 | 0.079 | 5.02 | 2.39 | 2.52 | 2.52 | 102.6 | 515.3 | 12.77 |

13.00 | 0.073 | 5.18 | 2.51 | 2.64 | 2.64 | 107.1 | 554.8 | 14.18 |

Speed (m/s) | t_{c}(s) | SR (steps/s) | SL (m) | SL/L_{0}(ratio) | F_{Zavg}/mg(ratio) | 𝜃_{total} (deg) | ω_{avg}(deg/s) | α_{max}(deg/s ^{2} × 10^{3}) |
---|---|---|---|---|---|---|---|---|

7.00 | 0.150 | 3.70 | 1.89 | 1.80 | 1.80 | 78.3 | 290.1 | 5.30 |

8.00 | 0.131 | 3.98 | 2.01 | 1.91 | 1.91 | 82.0 | 326.4 | 6.41 |

9.00 | 0.117 | 4.23 | 2.13 | 2.03 | 2.03 | 85.8 | 362.6 | 7.56 |

10.00 | 0.105 | 4.44 | 2.25 | 2.14 | 2.14 | 89.7 | 398.5 | 8.74 |

11.00 | 0.095 | 4.64 | 2.37 | 2.26 | 2.26 | 93.6 | 434.4 | 9.95 |

12.00 | 0.088 | 4.82 | 2.49 | 2.37 | 2.37 | 97.6 | 470.2 | 11.18 |

13.00 | 0.081 | 4.98 | 2.61 | 2.49 | 2.49 | 101.6 | 505.9 | 12.44 |

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Clark, K.P.
Determinants of Top Speed Sprinting: Minimum Requirements for Maximum Velocity. *Appl. Sci.* **2022**, *12*, 8289.
https://doi.org/10.3390/app12168289

**AMA Style**

Clark KP.
Determinants of Top Speed Sprinting: Minimum Requirements for Maximum Velocity. *Applied Sciences*. 2022; 12(16):8289.
https://doi.org/10.3390/app12168289

**Chicago/Turabian Style**

Clark, Kenneth P.
2022. "Determinants of Top Speed Sprinting: Minimum Requirements for Maximum Velocity" *Applied Sciences* 12, no. 16: 8289.
https://doi.org/10.3390/app12168289