Focus on Physics of Sport

Measurements are presented of the drag force on a golf ball dropped vertically into a tank of water. As observed previously in air, the drag coefficient drops sharply when the flow becomes turbulent. The experiment would be suitable for undergraduate students since it can be undertaken at low ball speeds and since the effects of turbulence are easily observed on video film. A modified golf ball was used to show how a ball with a smooth and a rough side, such as a cricket ball, is subject to a side force when the ball surface itself is asymmetrical in the transverse direction.

Croquet is a sport that is similar to billiards in that it involves the collision of one ball with another. Measurements and calculations are presented for three typical shots, one known as a straight croquet, one known as a split croquet and the other known as a push or roll shot. Each exhibit collision phenomena that could serve to illustrate aspects of an introductory course in mechanics, and that could also provide challenges even for more advanced students.

The kinematics of a free-kick is studied. As in projectile motion, the free-kick is ideal since we assume that a point-like ball moves in the absence of air resistance. We have experienced the fortunate conjuncture of a classical mechanics lecture taught right before an important football game. These types of sports events might trigger a great deal of attention from the classroom. The idealized problem is devised in such a way that students are eager to come to the end of the whole story.

The aerodynamic properties of a spinning table tennis ball were investigated using flight experiments. Using high-speed video cameras, the trajectory and rotation of an official ball (Nittaku 3-Star Premium), which was launched by a three rotor machine, were recorded. The drag and lift coefficients (C_D and C_L) were determined by analysing the video images. The measurements covered the speed and rotation range of typical table tennis shots in the form of the Reynolds number (Re) and dimensionless spin rate (SP), i.e. 3.0 × 10⁴ < Re < 9.0 × 10⁴ and 0 < SP < 1.0, and C_D and C_L were obtained as functions of Re and SP. We determined that the lift coefficient C_L is not a monotonically increasing function of SP. A deep valley of C_L was found around SP = 0.5, and the lift force exerted on a spinning ball almost vanished at Re = 9.0 × 10⁴ and 0.48 < SP < 0.5. These results qualitatively agree with the results from recent wind tunnel tests, but quantitative differences owing to the unsteady nature of the flight experiments remain. This anomaly in the lift coefficient should be called the 'lift crisis'.

In downhill alpine skiing, skiers often exceed speeds of 120 km h⁻¹, with air resistance substantially affecting the overall race times. To date, studies on air resistance in alpine skiing have used wind tunnels and actual skiers to examine the relationship between the gliding posture and magnitude of drag and for the design of skiing equipment. However, these studies have not revealed the flow velocity distribution and vortex structure around the skier. In the present study, computational fluid dynamics are employed with the lattice Boltzmann method to derive the relationship between total drag and the flow velocity around a downhill skier in the full-tuck position. Furthermore, the flow around the downhill skier is visualised, and its vortex structure is examined. The results show that the total drag force in the downhill skier model is 27.0 N at a flow velocity of 15 m s⁻¹, increasing to 185.8 N at 40 m s⁻¹. From analysis of the drag distribution and the flow profile, the head, upper arms, lower legs, and thighs (including buttocks) are identified as the major sources of drag on a downhill skier. Based on these results, the design of suits and equipment for reducing the drag from each location should be the focus of research and development in ski equipment. This paper describes a pilot study that introduces undergraduate students of physics or engineering into this research field. The results of this study are easy to understand for undergraduate students.

The aim was to run a case study of the biomechanics of a wheelchair sprinter racing the 100 m final at the 2016 Paralympic Games. Stroke kinematics was measured by video analysis in each 20 m split. Race kinetics was estimated by employing an analytical model that encompasses the computation of the rolling friction, drag, energy output and energy input. A maximal average speed of 6.97 m s⁻¹ was reached in the last split. It was estimated that the contributions of the rolling friction and drag force would account for 54% and 46% of the total resistance at maximal speed, respectively. Energy input and output increased over the event. However, we failed to note a steady state or any impairment of the energy input and output in the last few metres of the race. Data suggest that the 100 m is too short an event for the sprinter to be able to achieve his maximal power in such a distance.

Measurements are presented on the drag and lift coefficients for three relatively smooth balls launched in air and tracked with two cameras separated horizontally by 6.4 m. The ball spin was varied in order to investigate whether the Magnus force would increase or decrease when the ball spin was increased. For one ball, the Magnus force increased. For another ball, the Magnus force decreased almost to zero after reaching a maximum. For the third ball, the Magnus force was negative at low ball spins and positive at high ball spins. For one of the balls, the ball spin increased with time as it travelled through the air.

Trajectory analysis is an alternative to using wind tunnels to measure a soccer ball's aerodynamic properties. It has advantages over wind tunnel testing such as being more representative of game play. However, previous work has not presented a method that produces complete, speed-dependent drag and lift coefficients. Four high-speed cameras in stereo-calibrated pairs were used to measure the spatial co-ordinates for 29 separate soccer trajectories. Those trajectories span a range of launch speeds from 9.3 to 29.9 m s⁻¹. That range encompasses low-speed laminar flow of air over a soccer ball, through the drag crises where air flow is both laminar and turbulent, and up to high-speed turbulent air flow. Results from trajectory analysis were combined to give speed-dependent drag and lift coefficient curves for the entire range of speeds found in the 29 trajectories. The average root mean square error between the measured and modelled trajectory was 0.028 m horizontally and 0.034 m vertically. The drag and lift crises can be observed in the plots of drag and lift coefficients respectively.

Despite the impressiveness of the sprints run by Usain Bolt, the question naturally arises of why he has not been able to break the 100 m sprint world record he set in Berlin (2009). In this paper, we address such a query by considering Bolt's condition and the prevailing circumstances during the sprints that took place in Beijing 2008, Berlin 2009, London 2012, Moscow 2013, Beijing 2015 and Rio 2016³. Using the analytical mechanical model by Hernández-Gómez et al (2013), we analyse all the events, equating what we thought were the principal factors a priori: tailwind, weight gain and age. Despite what one might expect about the role of age in such a high-performance athlete as Usain Bolt, our results show that his performance has been essentially constant from Beijing 2009 to Rio 2016, with the mass gain and tailwind conditions making the difference in the run times he has achieved since Berlin 2009. Actually, our analysis suggests that in equal mass and tailwind conditions, his world record could actually have been set at Beijing 2015.

The aerodynamic properties of an arrow (A/C/E; Easton) were investigated in an extension of our previous work, in which the laminar-turbulent transition of the boundary layer on the arrow shaft was found to take place in the Re number range of 1.2 × 10⁴ < Re < 2.0 × 10⁴. In this paper, we focus on the influence of the arrow's attitude on the transition. Two types of vane (Spin Wing vane and Gas Pro vane) are fletched, and their stabilizing effects are compared. Two support-interference-free tests are performed to provide aerodynamic properties such as the drag, lift and pitching moment coefficients. The static aerodynamic properties are measured in a wind tunnel with JAXA's 60 cm magnetic suspension and balance system. When the arrow is aligned with the flow, the boundary layer remains laminar for Re < 1.5 × 10⁴, and the drag coefficient is approximately 1.5 for 1.0 × 10⁴ < Re < 1.5 × 10⁴. If the arrow has an angle of attack of 0.75 ° with the flow, the transition to turbulence takes place at approximately Re = 1.1 × 10⁴, and the drag coefficient increases to approximately 3.1. In addition, free flight experiments are performed. The arrow's velocity and angular velocity are recorded using five high-speed video cameras. By analysing the recorded images, we obtain the initial and final velocities from which the drag coefficient is determined. The trajectory and attitude of the arrow in free flight are computed numerically by integrating the equations of motion for a rigid body using the initial data obtained from the video images. The laminar-turbulent transition of the boundary layer is shown to take place, if the maximum angle of attack exceeds about 0.4° at Re = 1.75 × 10⁴. The crucial influence of the initial angular velocity on the angle of attack is also examined.

Slacklining is a new, rapidly expanding sport, and understanding its physics is paramount for maximizing fun and safety. Yet, compared to other sports, very little has been published so far on slackline dynamics. The equations of motion describing a slackline are fundamentally nonlinear, and assuming linear elasticity, they lead to a form of the Duffing equation. Following this approach, characteristic examples of slackline motion are simulated, including trickline bouncing, leash falls and longline surfing. The time-dependent solutions of the differential equations describing the system are acquired by numerical integration. A simple form of energy dissipation (linear drag) is added in some cases. It is recognized in this study that geometric nonlinearity is a fundamental aspect characterizing the dynamics of slacklines. Sports, and particularly slackline, is an excellent way of engaging young people with physics. A slackline is a simple yet insightful example of a nonlinear oscillator. It is very easy to model in the laboratory, as well as to rig and try on a university campus. For instructive purposes, its behaviour can be explored by numerically integrating the respective equations of motion. A form of the Duffing equation emerges naturally in the analysis and provides a powerful introduction to nonlinear dynamics. The material is suitable for graduate students and undergraduates with a background in classical mechanics and differential equations.

Drag and lift coefficients of recent FIFA world cup balls are examined. We fit a novel functional form to drag coefficient curves and in the absence of empirical data provide estimates of lift coefficient behaviour via a consideration of the physics of the boundary layer. Differences in both these coefficients for recent balls, which result from surface texture modification, can significantly alter trajectories. Numerical simulations are used to quantify the effect these changes have on the flight paths of various balls. Altitude and temperature variations at recent world cup events are also discussed. We conclude by quantifying the influence these variations have on the three most recent world cup balls, the Brazuca, the Jabulani and the Teamgeist. While our paper presents findings of interest to the professional sports scientist, it remains accessible to students at the undergraduate level.

The main contribution of this paper is didactic adaptation of the biomechanical analysis of the three main lifts in powerlifting (squat, bench press, deadlift). We used simple models that can easily be understood by undergraduate college students to estimate the values of various physical quantities during powerlifting. Specifically, we showed how plate choice affects the bench press and estimated spine loads and torques at hip and knee during lifting. Theoretical calculations showed good agreement with experimental data, proving that the models are valid.

The kinematics and dynamics of projectiles in sports is a complex topic involving several physical quantities and variables such as time, distance, velocity, acceleration, momentum, force, energy, viscosity, pressure, torque, bounce, sliding, rolling, etc. The analysis of these complex sets of multidimensional information, including the correlation between different variables, is an important requirement for the clear understanding of projectile trajectories in sports. However, those who do not have a strong mechanics or physics background find it difficult to interpret the data and comprehend the results in terms of the interacting forces and mutual interaction, which perpetuate the motion of the ball (or projectile). To address this issue, we propose a novel multivariate-data-visualization-based understanding of projectiles in sports inspired by the basic Gestalt principle that the whole is greater than the sum of its parts. The data representation approach involves the use of a single two-dimensional plane for the representation of multidimensional dynamic variables, and thereby completely removes the requirement of using several 2D plots for analysing and comprehending the meaning behind all of the data and how it correlates. For this study, we have considered the dynamics of two ball sports, namely volleyball and table tennis, as well as the sport of badminton, which involves high-drag projectile motion. We have presented a basic computational model incorporating the important forces to study projectile motion in sports. The data generated by the simulation is investigated using the proposed visualization methodology, and we show how this helps it to be interpreted easily, improving the clarity of our understanding of projectile trajectories in sports using both force and energy language.

This article is to review, for the benefit of university teachers, the most important arguments concerning the theory of sailing, especially regarding its high-speed aspect. The matter presented should be appropriate for students with basic knowledge of physics, such as advanced undergraduate or graduate. It is intended, furthermore, to put recent developments in the art of sailing in the proper historic perspective. We first regard the general geometric and dynamic conditions for steady sailing on a given course and then take a closer look at the high-speed case and its counter-intuitive aspects. A short overview is given on how the aero-hydrodynamic lift force arises, disposing of some wrong but entrenched ideas. The multi-faceted, composite nature of the drag force is expounded, with the special case of wave drag as a phenomenon at the boundary between different media. It is discussed how these various factors have to contribute in order to attain maximum speed. Modern solutions to this optimisation problem are considered, as well as their repercussions on the sport of sailing now and in the future.

Our daily activities involve different types of locomotion, including walking and running. In this paper we present a series of biomechanical models to describe such locomotor activities with a set only a few characteristic parameters. With this we aim at providing an introduction to the modelling of human locomotion which is equally appropriate for high-school and university-level students. We share our experiences with integrating these biomechanical models into classes with students of different disciplines comprising engineering, computer science, physics and sport science. Supplementing these modelling approaches, we use experimental methods as well as robotic testbeds to evaluate and justify the biomechanical simulation models. By this, we enable students from different fields to gain first experiences in the understanding and application of human movement systems and potential extensions to related technologies, such as robotics and assistive devices (e.g. prostheses and exoskeletons).