Next Article in Journal
Ear Centering for Accurate Synthesis of Near-Field Head-Related Transfer Functions
Next Article in Special Issue
Inter-Session Repeatability of Marker-Less Motion Capture of Treadmill Running Gait
Previous Article in Journal
Determining the Best Dressing Parameters for External Cylindrical Grinding Using MABAC Method
Previous Article in Special Issue
Do Carbon-Plated Running Shoes with Different Characteristics Influence Physiological and Biomechanical Variables during a 10 km Treadmill Run?
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Communication

Determinants of Top Speed Sprinting: Minimum Requirements for Maximum Velocity

Department of Kinesiology, West Chester University of Pennsylvania, West Chester, PA 19383, USA
Appl. Sci. 2022, 12(16), 8289; https://doi.org/10.3390/app12168289
Submission received: 28 June 2022 / Revised: 13 August 2022 / Accepted: 17 August 2022 / Published: 19 August 2022
(This article belongs to the Special Issue Running Biomechanics: From Commuting to Elite)

Abstract

:
Faster top sprinting speeds require shorter ground contact times, larger vertical forces, and greater thigh angular velocities and accelerations. Here, a framework using fundamental kinematic and kinetic relationships is presented that explores the effect of body dimensions on these mechanical determinants of sprinting performance. The analysis is applied to three hypothetical runners of different leg lengths to illustrate how these mechanical determinants of speed vary with body dimensions. Specific attention is focused on how the following variables scale with leg length and top speed: ground contact time, step rate, step length, ratio of step length to leg length, ratio of vertical force to body weight, total thigh range of motion, average thigh angular velocity, and maximum thigh angular acceleration. The analysis highlights the inherent biological tradeoffs that interplay to govern the optimal dimensions for sprinting speed and underscores that accounting for leg length may facilitate interpretation in future investigations examining the relationship between these mechanical variables and top speed. Furthermore, for athletes with given body dimensions and sprinting performance goals, this framework could help to establish the minimum requirements for maximum velocity.

1. Introduction

Research in locomotor biomechanics has increasingly elucidated the kinematic and kinetic mechanisms underpinning high-speed running performance. During upright running, ground contact time decreases with increasing speed [1,2,3], demanding greater mass-specific vertical forces to meet the necessary vertical impulse requirements [2,3,4]. Additionally, numerous publications have established that limb angular motion is an important factor in determining running speed [5,6,7,8]. Specifically, recent research has demonstrated that thigh angular velocity and thigh angular acceleration have a strong linear correlation to top speed [5,9].
Body dimensions (and particularly leg length and body mass) affect the kinetic and kinematic requirements for running at a given steady speed, whether examining human sprinters of different dimensions [8,10,11] or comparing various species of bipedal runners [12]. Based on limb excursion angles normally selected during ground contact [12,13,14], the constraints imposed by leg length are especially relevant to the factors of ground contact time, vertical impulse, and thigh angular motion. Therefore, a specific analysis of how body dimensions affect these factors is warranted.
Here, a simple framework utilizing basic kinematic and kinetic relationships is presented, in order to predict output variables observed from human runners when sprinting. The relationship of these variables represents the theoretical requirements for attaining maximum velocity across a range of body dimensions and top speeds. This analysis may provide additional insight into the mechanics underlying high-speed running, and potentially serve as a point of comparison for practitioners working with athletes.

2. Materials and Methods

The framework presented here enables the evaluation and prediction of kinematic and kinetic variables from top-speed sprinting by simply using the input parameters of running speed and leg length.
During steady-speed running, the center of mass horizontal velocity during the flight phase is constant (discounting resistance due to air friction), and so the runner’s forward Speed (m/s) is determined by the time it takes the center of mass to traverse the contact length Lc (m) during the ground contact time tc (s) [3,15]:
S p e e d = L c t c  
Lc is determined by leg length L0 (m) and the total excursion angle during contact θc (deg):
  L c = 2   · L 0   ·   sin   θ C 2    
For faster running speeds, prior research has demonstrated that θc ≈ 60 deg for humans and other bipedal runners [12,13,14]. Based on Equation (2), it can be approximated that LcL0 (see Figure 6 in [5]). Equation (1) then shows that tc is directly related to L0 and Speed:
t c = L 0 S p e e d    
Aerial time ta (s) is the time interval from takeoff of one foot until touchdown of the contralateral foot. Prior research has demonstrated that ta 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 tstep (s) is the time interval to complete one step, and is equal to the sum of tc and ta:
t s t e p = t c + t a  
Step Rate SR (steps/s) is the number of steps taken per second and is the inverse of tstep:
S R = 1 t s t e p    
Step Length SL (m) is the distance traveled per step and can be calculated from Speed and SR:
S L = S p e e d S R
During steady-speed running where the net vertical displacement of the body is zero, the time-averaged vertical ground reaction force must equal the body’s weight. Thus, the ratio of stance-averaged vertical force to body weight (FZavg/mg) can be determined if tc and ta are known:
F Z a v g m g = t s t e p t c = ( 1 + t a t c )  
where m (kg) is body mass and g (9.8 m/s2) is gravitational acceleration [16,17]. Given that tc is generally inversely related to top speed, and ta remains relatively constant across a range of runners and top speeds, FZavg/mg generally increases with top speed [2,3,4,17].
Limb angular motion is also considered within the context of this framework. Figure 1a depicts a simplified example of harmonic oscillatory thigh motion that assumes symmetrical peak thigh flexion and extension values. Because this framework is based on equations of harmonic motion [18], thigh angular kinematics as a function of time are determined by the parameters of frequency and θtotal, 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 tstep:
f = 1 T = 1 2 · t s t e p    
The angular position is referenced to the condition θ(t) = 0 at t = 0, and the equation for thigh angular position θ(t) is:
θ ( t ) = θ t o t a l 2   ·   sin ( 2 · π · f · t )    
The relationship between total thigh excursion θtotal (deg) and the total excursion angle during contact θc (deg) can then be derived from Equation (9) using the conditions θ(t) = θc/2 at t = T/2 − tc/2, and T = 1/f:
θ t o t a l = 2 · θ ( t ) sin ( 2 · π · f · t ) = θ c sin ( π · f · t c )  
The thigh angular velocity averaged throughout the stride cycle ωavg (deg/s) has been shown to increase linearly with top speed [5] and can be determined by:
ω a v g = θ t o t a l t s t e p = θ t o t a l   ·   S R    
The modeled maximum thigh angular acceleration αmax (deg/s2) 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:
α m a x = θ t o t a l 2 · ( 2 · π · f ) 2 = 2 · π 2 · θ t o t a l · f 2  
Collectively, the figures and equations presented above allow for predictions of key kinematic and kinetic variables across a range of top speeds. In Table 1, Table 2 and Table 3, values are calculated across a range of top speeds from 7.0 to 13.0 m/s for three hypothetical runners of different leg lengths: L0 = 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, θc = 60 deg and ta = 0.12 s.

3. Results

Data across a range of top speeds for the three hypothetical runners of varying leg lengths are presented in Table 1, Table 2 and Table 3 and Figure 2. These values are calculated from the equations and simplifying assumptions presented in the Methods. The predictive lines illustrated in Figure 2 are based on the outcomes listed in Table 1, Table 2 and Table 3.
The calculated outcomes in Table 1, Table 2 and Table 3 include values for the following variables during top speed sprinting: running Speed, ground contact time (tc), step rate (SR), step length (SL), ratio of step length to leg length (SL/L0), ratio of stance-averaged vertical force to body weight (FZavg/mg), total thigh range of motion from peak extension to peak flexion (θtotal), thigh angular velocity averaged throughout stride cycle (ωavg), and modeled maximum thigh angular acceleration during the swing phase (αmax).

4. Discussion

As demonstrated in Table 1, Table 2 and Table 3 and Figure 2, several basic relationships were observed across the range of top speeds for the hypothetical runners of different body dimensions. There was a negative relationship between Speed and tc (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 FZavg/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 θtotal, ωavg, and αmax (Figure 2f–h) aligning with recent experimental data on thigh angular motion at top speed [5,9].
As illustrated in Figure 2a–h, the value of L0 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 tc; (B) decreased SR; (C) longer SL; (D) decreased SL/L0 ratio; (E) reduced FZavg/mg; (F) decreased θtotal; (G) slower ωavg; and (H) decreased αmax. 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 (mL02), 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.
To enable insight into the above variables, this framework included two simplifying kinematic assumptions based on prior research: θc = 60 deg and ta = 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 θc = 60 deg will affect the other calculated variables, with increased θc allowing for longer tc and decreased FZavg/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 ta = 0.12 ± 0.02 s [2,3,4,5], and thus employing a standard ta here for hypothetical runners with varying L0 likely did not introduce major errors. However, differences in individual running styles may exist [20,21] that could result in ta outside the standard range presented here, subsequently affecting the other calculated variables.
In spite of the simplifying assumptions included in this theoretical approach, several important insights are highlighted by the calculated outcome variables. First, future investigations examining the determinants of top speed sprinting may aim to statistically account for leg length in order to properly analyze the relationship of mechanical variables to performance. Although this has been done in some recent investigations into human sprinting performance [8], as well as in research on bipedal locomotion across species [12], the results here indicate just how substantial the effect of leg length is on speed-specific requirements. Furthermore, the data presented here could provide normative ranges for the mechanical variables needed to attain a specific top speed for runners of different body dimensions. While further direct experimental investigation is needed to validate these kinematic variables in relation to L0 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

During top-speed sprinting, body dimensions interplay with the mechanical variables of ground contact time, step rate and length, vertical force application, and leg angular motion to determine performance. Here, a simple framework was presented to analyze these variables within the context of running speed and leg length. The results of this investigation demonstrate that accounting for leg length may facilitate interpretation when analyzing the relationship of mechanical variables to top-speed sprinting performance. Furthermore, when evaluating athletes with given body dimensions and sprinting performance goals, this framework may help to establish the minimum requirements for maximum velocity.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The author thanks Laurence Ryan for his helpful input on this manuscript.

Conflicts of Interest

The author declares no conflict of interest.

List of Symbols and Abbreviations

θ(t)thigh angle as a function of time (deg)
θctotal thigh excursion during the ground contact phase (deg)
θtotaltotal thigh range of motion from peak extension to peak flexion (deg)
ωavgthigh angular velocity averaged throughout stride cycle (deg/s)
αmaxmodeled maximum thigh angular acceleration during swing (deg/s2)
FZavg/mgratio of stance-averaged vertical force to body weight mg
ffrequency of thigh angular motion (Hz)
ggravitational acceleration (9.8 m/s2)
Lccontact length (m)
L0leg length (m)
mbody mass (kg)
SLstep length (m)
SL/L0ratio of step length to leg length
Speedrunner’s forward speed
SRstep rate (steps/s)
Ttime period of thigh angular motion (s)
ttime (s)
taaerial time (s)
tcground contact time (s)
tstepstep time (s)

References

  1. Nummela, A.; Keränen, T.; Mikkelsson, L.O. Factors related to top running speed and economy. Int. J. Sports Med. 2007, 28, 655–661. [Google Scholar] [CrossRef] [PubMed]
  2. Weyand, P.G.; Sternlight, D.B.; Bellizzi, M.J.; Wright, S. Faster top running speeds are achieved with greater ground forces not more rapid leg movements. J. Appl. Physiol. 2000, 89, 1991–1999. [Google Scholar] [CrossRef] [PubMed]
  3. Weyand, P.G.; Sandell, R.F.; Prime, D.N.; Bundle, M.W. The biological limits to running speed are imposed from the ground up. J. Appl. Physiol. 2010, 108, 950–961. [Google Scholar] [CrossRef] [PubMed]
  4. Clark, K.P.; Weyand, P.G. Are running speeds maximized with simple-spring stance mechanics? J. Appl. Physiol. 2014, 117, 604–615. [Google Scholar] [CrossRef] [PubMed]
  5. Clark, K.P.; Meng, C.R.; Stearne, D.J. ‘Whip from the hip’: Thigh angular motion, ground contact mechanics, and running speed. Biol. Open 2020, 9, bio053546. [Google Scholar] [CrossRef] [PubMed]
  6. Kivi, D.M.; Maraj, B.K.; Gervais, P. A kinematic analysis of high-speed treadmill sprinting over a range of velocities. Med. Sci. Sports Exerc. 2002, 34, 662–666. [Google Scholar] [CrossRef] [PubMed]
  7. Mann, R.V.; Murphy, A. The Mechanics of Sprinting and Hurdling; CreateSpace Independent Publishing Platform: Las Vegas, NV, USA, 2018; pp. 155–157. [Google Scholar]
  8. Miyashiro, K.; Nagahara, R.; Yamamoto, K.; Nishijima, T. Kinematics of maximal speed sprinting with different running speed, leg length and step characteristics. Front. Sports Act. Liv. 2019, 1, 37. [Google Scholar] [CrossRef] [PubMed]
  9. Clark, K.P.; Ryan, L.J.; Meng, C.R.; Stearne, D.J. Evaluation of maximum thigh angular acceleration during the swing phase of steady-speed running. Sports Biomech. 2021; published online ahead of print. [Google Scholar]
  10. Beneke, R.; Taylor, M.J. What gives Bolt the edge—AV Hill knew it already! J. Biomech. 2010, 43, 2241–2243. [Google Scholar] [CrossRef] [PubMed]
  11. Weyand, P.G.; Davis, J.A. Running performance has a structural basis. J. Exp. Biol. 2005, 208, 2625–2631. [Google Scholar] [CrossRef] [PubMed]
  12. Gatesy, S.M.; Biewener, A.A. Bipedal locomotion: Effects of speed, size and limb posture in birds and humans. J. Zool. 1991, 224, 127–147. [Google Scholar] [CrossRef]
  13. Farley, C.T.; Glasheen, J.; McMahon, T.A. Running springs: Speed and animal size. J. Exp. Biol. 1993, 185, 71–86. [Google Scholar] [CrossRef] [PubMed]
  14. He, J.P.; Kram, R.; McMahon, T.A. Mechanics of running under simulated low gravity. J. Appl. Physiol. 1991, 71, 863–870. [Google Scholar] [CrossRef] [PubMed]
  15. Cavagna, G.A.; Thys, H.; Zamboni, A. The sources of external work in level walking and running. J. Physiol. 1976, 262, 639–657. [Google Scholar] [CrossRef]
  16. Clark, K.P.; Ryan, L.J.; Weyand, P.G. Foot speed, foot-strike and footwear: Linking gait mechanics and running ground reaction forces. J. Exp. Biol. 2014, 217, 2037–2040. [Google Scholar] [CrossRef]
  17. Clark, K.P.; Ryan, L.J.; Weyand, P.G. A general relationship links gait mechanics and running ground reaction forces. J. Exp. Biol. 2017, 220, 247–258. [Google Scholar] [CrossRef]
  18. McGeer, T. Passive bipedal running. Proc. Roy. Soc. Lon. B Biol. Sci. 1990, 240, 107–134. [Google Scholar] [CrossRef]
  19. Biewener, A.A. Scaling body support in mammals: Limb posture and muscle mechanics. Science 1989, 245, 45–48. [Google Scholar] [CrossRef]
  20. Gindre, C.; Lussiana, T.; Hebert-Losier, K.; Mourot, L. Aerial and terrestrial patterns: A novel approach to analyzing human running. Int. J. Sports Med. 2016, 37, 25–26. [Google Scholar] [CrossRef]
  21. van Oeveren, B.T.; de Ruiter, C.J.; Beek, P.J.; van Dieën, J.H. The biomechanics of running and running styles: A synthesis. Sports Biomech. 2021; published online ahead of print. [Google Scholar] [CrossRef]
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 θtotal, which is the total thigh range of motion from peak extension through peak flexion. The total excursion angle during contact (θc) occurs in the contact time (tc) between t = T/2 − tc/2 and t = T/2 + tc/2.
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 θtotal, which is the total thigh range of motion from peak extension through peak flexion. The total excursion angle during contact (θc) occurs in the contact time (tc) between t = T/2 − tc/2 and t = T/2 + tc/2.
Applsci 12 08289 g001
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 (L0). (a) ground contact time (tc), (b) step rate (SR), (c) step length (SL), (d) ratio of step length to leg length (SL/L0), (e) ratio of stance-averaged vertical force to body weight (FZavg/mg), (f) total thigh range of motion (θtotal), (g) average thigh angular velocity (ωavg), (h) modeled maximum thigh angular acceleration (αmax).
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 (L0). (a) ground contact time (tc), (b) step rate (SR), (c) step length (SL), (d) ratio of step length to leg length (SL/L0), (e) ratio of stance-averaged vertical force to body weight (FZavg/mg), (f) total thigh range of motion (θtotal), (g) average thigh angular velocity (ωavg), (h) modeled maximum thigh angular acceleration (αmax).
Applsci 12 08289 g002
Table 1. Calculated values across a range of speeds for a hypothetical athlete with L0 = 0.85 m.
Table 1. Calculated values across a range of speeds for a hypothetical athlete with L0 = 0.85 m.
Speed
(m/s)
tc
(s)
SR
(steps/s)
SL
(m)
SL/L0
(ratio)
FZavg/mg
(ratio)
𝜃total
(deg)
ωavg
(deg/s)
αmax
(deg/s2 × 103)
7.000.1214.141.691.991.9984.5349.87.15
8.000.1064.421.812.132.1389.2394.38.60
9.000.0944.661.932.272.2794.1438.610.09
10.000.0854.882.052.412.4199.0482.811.62
11.000.0775.072.172.552.55104.0526.913.18
12.000.0715.242.292.692.69109.0571.114.77
13.000.0655.392.412.842.84114.0615.216.38
Table 2. Calculated values across a range of speeds for a hypothetical athlete with L0 = 0.95 m.
Table 2. Calculated values across a range of speeds for a hypothetical athlete with L0 = 0.95 m.
Speed
(m/s)
tc
(s)
SR
(steps/s)
SL
(m)
SL/L0
(ratio)
FZavg/mg
(ratio)
𝜃total
(deg)
ωavg
(deg/s)
αmax
(deg/s2 × 103)
7.000.1363.911.791.881.8881.0316.96.12
8.000.1194.191.912.012.0185.2356.97.38
9.000.1064.432.032.142.1489.5396.68.68
10.000.0954.652.152.262.2693.8436.310.01
11.000.0864.852.272.392.3998.2475.811.38
12.000.0795.022.392.522.52102.6515.312.77
13.000.0735.182.512.642.64107.1554.814.18
Table 3. Calculated values across a range of speeds for a hypothetical athlete with L0 = 1.05 m.
Table 3. Calculated values across a range of speeds for a hypothetical athlete with L0 = 1.05 m.
Speed
(m/s)
tc
(s)
SR
(steps/s)
SL
(m)
SL/L0
(ratio)
FZavg/mg
(ratio)
𝜃total
(deg)
ωavg
(deg/s)
αmax
(deg/s2 × 103)
7.000.1503.701.891.801.8078.3290.15.30
8.000.1313.982.011.911.9182.0326.46.41
9.000.1174.232.132.032.0385.8362.67.56
10.000.1054.442.252.142.1489.7398.58.74
11.000.0954.642.372.262.2693.6434.49.95
12.000.0884.822.492.372.3797.6470.211.18
13.000.0814.982.612.492.49101.6505.912.44
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

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

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop