Kinesiology and Biomechanics II: the Laws of Motion and their Application in Sports, Projectiles and the Principles of Projection, Linear and Angular Kinematics and Kinetics, Friction, Spin, Impact and Elasticity, and Air and Water Dynamics

A projectile is any body released into the air and then acted on only by gravity and air resistance: a shot, a javelin, a ball, or a jumper's own body after take-off. Ignoring air resistance, its horizontal velocity stays constant while its vertical velocity changes at 9.8 m/s² (often taken as 10 m/s² for calculation), so its path is a parabola. The flight of the centre of gravity cannot be changed once the body is airborne; a jumper can only rearrange the limbs around it. Three release factors decide the result: the speed of release, the angle of release and the height of release, and of these speed matters most, because the range depends on its square.

When release and landing are at the same height, the horizontal range is R = v² sin 2θ ÷ g, the time of flight T = 2v sin θ ÷ g and the maximum height H = v² sin²θ ÷ 2g. Because sin 2θ is greatest (equal to 1) when 2θ = 90°, the range is greatest at θ = 45°, and complementary angles such as 30° and 60° or 15° and 75° give equal ranges. Worked examples with g = 10 m/s²: at 10 m/s and 45°, R = 100 × 1 ÷ 10 = 10 m; at 20 m/s and 15°, R = 400 × sin 30° ÷ 10 = 400 × 0.5 ÷ 10 = 20 m; at 20 m/s and 30°, T = 2 × 20 × 0.5 ÷ 10 = 2 s and H = 400 × 0.25 ÷ 20 = 5 m. Doubling the release speed quadruples the range.

Real athletes release well below 45°, for three reasons. First, the release height is above the landing height: a shot is released at about 2 m and lands on the ground, so the projectile has extra time to travel and the optimum angle falls below 45°, the more so the greater the release height relative to the distance thrown. Second, and most important, release speed is not independent of release angle: the human body can generate greater speed when projecting more horizontally, because the muscles and joints are better placed to push forward than up and less work is done against gravity during the delivery; so a slightly lower angle with a higher speed beats 45° with a lower speed. Third, aerodynamics: a javelin and a discus generate lift and drag, which change the optimum. The result is that elite shot-putters release at roughly 35–38°, javelin throwers in the low to mid 30s, and long jumpers take off at only about 18–25°, because a jumper running at around 10 m/s cannot convert that horizontal speed into a steep take-off without losing most of it. A basketball shot needs a steeper arc to enter the ring, and a ski jumper or a discus thrower works within aerodynamic forces as much as gravity.

Linear motion (translation) is motion in which all parts of a body move the same distance in the same direction: a skater gliding, a cyclist's trunk. Angular (rotary) motion is motion about an axis: a gymnast on the bar, the leg swinging at the hip. Most sporting movement is general motion, a combination, in which angular motions of the limbs produce linear motion of the body or an implement. Linear kinematics uses distance and displacement, speed and velocity, and acceleration; running speed, for example, equals stride length × stride frequency. Angular kinematics uses angular displacement (degrees or radians; 360° = 2π rad), angular velocity ω = angular displacement ÷ time and angular acceleration. A diver who completes two and a half somersaults (900°) in 1.5 s has an average angular velocity of 600°/s. The linear velocity of a point on a rotating body is v = ωr, with ω in radians per second: a golf club head 1.5 m from the axis of a swing turning at 30 rad/s moves at 45 m/s, which is why a longer lever at the same angular velocity strikes faster.

Angular kinetics is the study of what causes rotation. Torque (moment of force) = force × perpendicular distance from the axis; a force through the axis produces no rotation. The angular equivalent of mass is the moment of inertia, I = Σmr², which depends on how the mass is distributed about the axis: the farther the mass from the axis, the greater the resistance to rotation. Angular momentum L = Iω, and when no external torque acts, as in flight, it is conserved. So a diver or gymnast who tucks tightly reduces I and spins faster, then opens out to slow down before entry or landing; a figure skater pulls the arms in to spin faster; and a sprinter flexes the recovery leg to reduce its moment of inertia about the hip. Newton's laws have angular analogues: a body keeps rotating with constant angular momentum unless a torque acts; torque = I × angular acceleration; and every torque has an equal and opposite torque, as when a long jumper's arm circling in the air counter-rotates the trunk.

Friction is the force that opposes the sliding of one surface over another. Its magnitude is F = μ × R, where R is the normal reaction force pressing the surfaces together and μ is the coefficient of friction, which depends on the two surfaces; a force of 300 N needed to start sliding a 600 N load means μ = 0.5. Static friction, before sliding begins, is greater than kinetic (sliding) friction once it has begun, and rolling friction is smaller still. Sport needs friction in some places and minimises it in others: spikes, studs, rubber soles, chalk on a gymnast's hands and grips on rackets increase it; wax on skis, smooth ice for skating and ball bearings in cycles reduce it.

Spin changes both the flight and the bounce of a ball. In flight, a spinning ball drags a thin layer of air round with it, so the air speed and pressure differ on its two sides and a sideways force, the Magnus force, acts on it. Topspin (the top of the ball rotating forward) produces a downward force, so the ball dips sharply, as in a tennis topspin drive or a football shot that dips under the bar; backspin produces lift, so the ball floats and travels farther, as with a golf drive or a backspun volleyball float; sidespin curves the ball sideways, as in a curling free kick. On the bounce, a topspin ball skids forward low and fast, while a backspin ball bounces more steeply and checks. The swing of a new cricket ball is a different effect, produced by the seam and the difference in surface between the two sides rather than by spin.

Impact is the collision of two bodies over a very short time, and elasticity is a body's ability to regain its shape after deformation. The coefficient of restitution (e) measures the elasticity of an impact: e = relative velocity of separation ÷ relative velocity of approach, and for a ball dropped onto a fixed surface e = √(bounce height ÷ drop height). A ball dropped from 1.0 m that rebounds to 0.36 m has e = √0.36 = 0.6. e lies between 0 (perfectly inelastic, no bounce) and 1 (perfectly elastic); it depends on both ball and surface, rises with temperature (a warm squash ball bounces more) and falls with impact speed, and governing bodies test balls by drop-and-rebound to keep play consistent. A tennis ball on grass bounces lower and faster than on clay because grass has lower friction and absorbs energy differently.

Air and water are fluids and exert two kinds of dynamic force on a moving body: drag, which opposes motion, and lift, which acts perpendicular to the flow. Drag has three components: surface (skin friction) drag, from the fluid rubbing along the surface, reduced by smooth skin suits and shaving; form (pressure) drag, from the pressure difference between the front and the turbulent wake behind, reduced by streamlining, the cyclist's crouch and the swimmer's horizontal body position; and wave drag, at the water surface in swimming, reduced by swimming underwater after starts and turns within the rules. Drag rises roughly with the square of velocity, which is why it matters so much in cycling, skiing and sprint swimming, and why cyclists draft behind one another. Lift arises when the flow is deflected, as over an inclined javelin, discus or ski jumper, and by Bernoulli's principle faster flow over one surface means lower pressure there. The dimples on a golf ball make the boundary layer turbulent so that it clings to the ball longer, shrinking the wake and reducing form drag.

In water, buoyancy also acts. By Archimedes' principle the upthrust on an immersed body equals the weight of fluid it displaces, so whether a person floats depends on body density: fat is less dense than water and muscle and bone more dense, so lean, muscular swimmers tend to sink and people with more body fat float more easily. The centre of buoyancy (the centre of the displaced water, near the chest because of the lungs) usually lies above and towards the head from the centre of gravity (near the hips), so the two forces form a couple that rotates the legs downwards, which is why beginners' legs sink and why a good kick and a head-down, hips-up position matter. Swimming propulsion comes from both drag and lift forces generated by the hands and feet, and Newton's third law applies throughout: the swimmer pushes water backwards to be driven forwards.