Option C: The physics of sports

AS Unit 4 Option C: The physics of sports Learners should be able to demonstrate and apply their knowledge and understanding of:
a)	How to use the centre of gravity to explain how stability and toppling is achieved in various sporting contexts
b)	How to use the principle of moments to determine forces within • various muscle systems in the human body and • other sporting contexts, for example, sailing
c)	How to use Newton’s 2 nd law in the form Ft mv mu = − in various sporting contexts
d)	The coefficient of restitution as [math]c = \frac{\text{Relative speed after collision}}{\text{Relative speed before collision}}[/math] And also use it in the form [math]e = \sqrt{\frac{h}{H}}[/math] Where h is the bounce height and H is the drop height
e)	What is meant by the moment of inertia of a body
f)	How to use equations to determine the moment of inertia, I, for example o A solid sphere [math]I = \frac{2}{5} m r^2[/math] o A thin spherical shell [math]I = \frac{2}{3} m r^2[/math] where m is the mass and r is the radius
g)	The idea that angular acceleration, a , is defined as the rate of change of angular velocity, ω, and how to use the equation [math]\alpha = \frac{\omega_2 – \omega_1}{t}[/math]
h)	The idea that torque, τ, is given as [math]\tau = I \alpha[/math]
i)	Angular momentum, L, is given as [math]L = Iω[/math] where is the angular velocity
j)	The principle of conservation of angular momentum and use it to solve problems in sporting contexts
k)	How to use the equation for the rotational kinetic energy, rotational [math]KE = \frac{1}{2} Iω^2[/math]
l)	How to use the principle of conservation of energy including the use of linear and rotational kinetic energy as well as gravitational and elastic potential energy in various sporting contexts
m)	How to use projectile motion theory in sporting contexts
n)	How to use Bernoulli’s equation [math]p = p_0 – \frac{1}{2} ρv^2[/math] in sporting contexts
o)	How to determine the magnitude of the drag force using [math]F_D = \frac{1}{2} ρv^2 AC_D[/math] where [math]C_D[/math] is the drag coefficient

a) Center of Gravity, Stability, and Toppling
⇒ Center of Gravity (CG) Concept
Definition:
The center of gravity is the point in an object where the total weight is considered to be concentrated. In any gravitational field, the CG is the average location of the weight distribution.
Figure 1 Center of gravity
⇒ Application to Stability and Toppling
Stability:
An athlete or object is stable if the vertical line through its center of gravity falls within its base of support. For example, a soccer player standing with feet apart has a wider base, which helps maintain stability.
Toppling:
When the center of gravity moves outside the base of support, the object or athlete becomes unstable and may topple. For instance, in gymnastics, a beam routine requires careful balance so that the gymnast’s CG stays over the narrow beam; if it shifts too far, she may fall.
Sporting Examples:
– High Jump:
Athletes tilt their bodies to raise their center of gravity effectively over the bar while maintaining balance.
– Cycling:
Cyclists lean into turns to shift their CG over the wheels, preventing toppling.
– Weightlifting:
A low center of gravity and a wide stance improve stability during heavy lifts.
b) Principle of Moments in Muscle Systems and Other Sporting Contexts
⇒ Principle of Moments (Torque)
Definition:
The principle of moments states that for an object in rotational equilibrium, the sum of clockwise moments about a pivot equals the sum of anticlockwise moments. A moment (torque) is the product of a force and its perpendicular distance from a pivot.
Figure 2 Principle of momentum
⇒ Application in Muscle Systems
Muscle Levers:
In the human body, bones act as levers and joints serves as pivots. Muscles generate forces that produce moments about joints. For example, in the forearm, the biceps apply force to lift a weight, and the torque they generate depends on the lever arm distance from the elbow joint.
Calculations:
Torque = Force × Lever Arm
This helps in determining the muscle force required to produce a desired movement.
⇒ Application in Other Sporting Contexts (e.g., Sailing)
Sailing:
The sails generate forces that produce moments around the keel of a boat. The distribution of these forces and the resulting torques determine the boat’s balance and ability to resist capsizing.
Optimization:
By analyzing the moments about the pivot (keel), designers can optimize sail shape and keel design to improve performance and stability.
c) Newton’s Second Law in Sport
⇒ Impulse-Momentum Formulation
Newton’s Second Law:
The law is often written as:
F = ma or F∆t = m∆v
Where F∆t is the impulse and m∆v is the change in momentum.
⇒ Sporting Applications:
Sprinting:
Athletes generate force over a short time interval (impulse) to accelerate from rest.
Figure 3 Sprinting
Ball Sports:
When a player kicks a ball, the force applied over the contact time changes the ball’s momentum.
Figure 4 Newton’s second law apply on Ball
Deceleration:
In sports like gymnastics, forces are applied to decelerate or change direction, where the impulse-momentum relationship is key.
Negative Sign in Impulse Equations:
Often, the negative sign indicates that the force is acting opposite to the direction of motion, as seen during deceleration or braking.
d) Coefficient of Restitution
⇒ Definition Using Relative Speed
Equation:
[math]c = \frac{\text{Relative speed after collision}}{\text{Relative speed before collision}}[/math]
– c = 1 indicates a perfectly elastic collision (no kinetic energy loss).
– c = 0 indicates a perfectly inelastic collision (maximum energy loss).
⇒ Alternate Formulation with Bounce Height
Suppose drop an object from a height H and it bounces to a height h then the velocity at which it makes contact with the ground is given by:
[math]v_{\text{initial}} = \sqrt{2gH}[/math]
Where the SUVAT [math]v^2 = u^2 + 2as[/math] was used. Similarly, we can use the height reached after bouncing to calculate the velocity of the object the instant it has bounced and begins to travel upwards. This means that in the SUVAT ‘v’ is zero and we are solving for ‘u’, which gives:
[math]v_{\text{after}} = \sqrt{2gH}[/math]
Hence
[math]e = \frac{v_{\text{after}}}{v_{\text{initial}}} = \sqrt{\frac{h}{H}}[/math]
– h is the bounce height.
– H is the drop height.
– This form is especially useful for determining how “bouncy” a ball is.
⇒ Sporting Applications:
Basketball or Tennis:
Knowing the coefficient of restitution helps in designing balls with the desired bounce characteristics.
Impact Testing:
It’s used in materials testing and sports equipment design.
e) Moment of Inertia
Definition:
The moment of inertia (I) is a measure of a body’s resistance to rotational acceleration about a given axis. It depends on both the total mass of the object and how that mass is distributed relative to the axis of rotation.
[math]\sum m_i r_i^2[/math]
Where [math]m_i[/math], is the ith mass element and [math]r_i[/math] is the distance of that element to the axis of rotation.
Calculating the moment of inertia for common objects usually involves integration, and often non-cartesian coordinates so it is out of the scope of this course to do so but common moments of inertia are:
Figure 5 Momentum of Inertia of different objects
f) Examples of Inertia:
Solid Sphere:
[math]I = \frac{2}{5} m r^2[/math]
Here, m is the mass and r is the radius of the sphere.
Thin Spherical Shell:
[math]I = \frac{2}{3} m r^2[/math]
Here, m is the mass and r is the radius of the spherical shell.
g) Angular Acceleration (α)
⇒ Definition:
Angular acceleration is the rate of change of angular velocity with time.
⇒ Formula:
[math]\alpha = \frac{\omega_2 – \omega_1}{t}[/math]
Where:
– [math]\omega_1[/math] and [math]\omega_2[/math] are the initial and final angular velocities (in radians per second), and
– t is the time over which the change occurs.
Figure 6 Angular acceleration
Torque (τ)
Definition:
Torque is the measure of the force that can cause an object to rotate about an axis. It is analogous to force in linear motion.
Relation to Moment of Inertia and Angular Acceleration:
Newton’s second law for rotation is expressed as:
[math]\tau = I \alpha[/math]
This equation shows that the torque required to produce a given angular acceleration is directly proportional to the moment of inertia of the object.
Figure 7 An object rotates about its axis

i) Angular Momentum and Its Conservation
⇒ Definition of Angular Momentum (L)
Angular momentum (L) is the rotational equivalent of linear momentum.
It is given by the equation:
L = Iω
Where:
– I is the moment of inertia (resistance to rotational motion).
– ω is the angular velocity (rate of rotation).
Figure 8 Angular momentum
⇒ Interpretation:
The larger the moment of inertia or angular velocity, the greater the angular momentum.
Angular momentum depends on how mass is distributed around the axis of rotation.
j) Principle of Conservation of Angular Momentum
⇒ Statement:
“The total angular momentum of a system remains constant if no external torque acts on it.”
⇒ Mathematically:
[math]L_{initial} = L_{final}[/math]
or
[math]I_1 ω_1 = I_2 ω_2[/math]
If the moment of inertia changes, the angular velocity must adjust to keep L
⇒ Examples in Sporting Contexts:
Figure Skating:
A skater spinning with arms extended has a high moment of inertia (I) and a low angular velocity (ω). When they pull their arms in, I decreases, so ω increases, making them spin faster.
Diving:
A diver tucking their body during a somersault reduces I, increasing rotation speed.
Gymnastics:
Gymnasts adjust their body position to change rotational speed while in the air.
k) Rotational Kinetic Energy
⇒ Definition:
Just as linear motion has kinetic energy ([math]KE = \frac{1}{2} mv^2[/math] ), rotational motion also has kinetic energy:
[math]KE_{\text{rotational}} = \frac{1}{2} I \omega^2[/math]
⇒ Meaning:
The faster an object rotates (ω increases), the more rotational kinetic energy it has.
The greater the moment of inertia (I), the more energy is required to rotate the object.
Figure 9 Rotational Kinetic energy
⇒ Example in Sports:
Cycling: A rotating bicycle wheel stores rotational kinetic energy.
Discus Throw: The energy needed to spin the discus depends on its moment of inertia and angular velocity.
l) Principle of Conservation of Energy in Rotational Motion
⇒ Total Mechanical Energy (TME) is conserved:
The sum of kinetic energy (both rotational and linear), gravitational potential energy, and elastic potential energy remains constant.
⇒ Formula:
[math]PE + KE_{\text{linear}} + KE_{\text{rotational}} = \text{constant}[/math]
Where:
– [math]PE = mgh[/math](Gravitational potential energy)
– [math]KE_{linear} = \frac{1}{2} mv^2[/math]
– [math]KE_{rotational} = \frac{1}{2} Iω^2[/math]
⇒ Examples in Sporting Contexts:
Pole Vaulting: Gravitational potential energy is converted into kinetic energy, allowing the athlete to reach a higher position.
Ski Jumping: As a skier moves down a ramp, potential energy converts into kinetic and rotational energy.
Basketball Spin Shot: The ball’s rotation affects its trajectory and bounce due to rotational kinetic energy.
m) Projectile Motion in Sports
⇒ Concept:
Projectile motion describes the motion of an object that is launched into the air and moves under the influence of gravity, with no significant air resistance (or with controlled resistance). The key parameters include:
– Initial Velocity ( [math]v_o[/math])
– Launch Angle (θ)
– Acceleration Due to Gravity (g)
Figure 10 Projectile motion in sports
⇒ Equations:
Horizontal Range:
[math]R = \frac{v_0^2 \sin 2\theta}{g}[/math]
Time of Flight:
[math]T = \frac{2 v_0 \sin \theta}{g}[/math]
Maximum Height:
[math]H = \frac{v_0^2 \sin^2 \theta}{2g}[/math]
⇒ Application in Sports:
Throwing and Kicking:
In sports like football (soccer), baseball, or javelin throw, athletes aim to maximize range or accuracy. They adjust the launch angle and speed to optimize distance.
Basketball:
The arc of a jump shot can be analyzed using projectile motion principles to achieve the desired trajectory for a successful shot.
Ski Jumping:
Athletes use projectile motion concepts to determine optimal takeoff angles and speeds to maximize jump distance.
n) Bernoulli’s Equation in Sports
⇒ Concept:
Bernoulli’s equation relates the pressure p and velocity v of a fluid (or air) along a streamline:
[math]p = p_0 – \frac{1}{2} ρv^2[/math]
Where:
– p is the local pressure,
– [math]p_o[/math] is the reference pressure (often taken far from the object),
– ρ is the fluid density,
– v is the fluid velocity.
Figure 11 Bernoulli’s principle
⇒ Application in Sports:
Ballistics and Curved Trajectories:
In sports like baseball, a curveball’s spin creates differences in airflow around the ball. Faster flow on one side reduces pressure (Bernoulli’s principle), causing the ball to curve.
Aerodynamics of Equipment:
Cyclists and skiers use aerodynamic designs to reduce air pressure drag. Bernoulli’s equation helps in understanding how streamlined shapes can lower pressure differences and improve performance.
Sailing:
The shape of the sail and its interaction with wind involve Bernoulli’s principle to generate lift, which contributes to the force that propels the boat.
o) Determining Drag Force in Sports
⇒ Drag Force Equation:
Drag force ( [math]F_D[/math]) opposes an object’s motion through a fluid (such as air) and is given by:
[math]F_D = \frac{1}{2} \rho v^2 A C_D[/math]
Where:
– ρ is the density of the fluid (air),
– v is the velocity of the object relative to the fluid,
– A is the cross-sectional area facing the flow,
- [math]C_D[/math] is the drag coefficient, which depends on the shape and surface roughness of the object.
Figure 12 Drag Force
⇒ Application in Sports:
Ball Sports:
In sports like soccer, cricket, or tennis, understanding drag is crucial to predicting the flight of the ball. Players and equipment designers consider [math]C_D[/math] to optimize ball shape for better control.
Cycling and Running:
Athletes work to reduce drag by adopting aerodynamic postures and wearing streamlined clothing. Lowering A (effective frontal area) and optimizing body shape (reducing [math]C_D[/math]) can significantly improve performance.
Skiing and Motorsport:
Vehicles and athletes aim to minimize drag for speed. The equation helps engineers design fairings, helmets, and suits that reduce air resistance.