B2.1 Newton's laws of motion
🏃♂ Movement & Forces
Body movement and sporting equipment motion are governed by Newton’s laws of motion
Understanding these laws is essential for performance analysis, injury reduction, and technique development
📐 Key Mechanical Terms
Scientific terms such as force, power, velocity, and energy have specific meanings
Knowing these definitions allows for accurate and informative analysis of human movement
📚 Importance of Mechanics
Mechanical science explains how all moving bodies perform actions
Emphasizes the foundational role of mechanics in understanding movement
🧠 Key Idea
You cannot analyse sport or physical activity effectively without understanding Newton’s laws
📘 Kinematics
Study of motion
Motion = change in position of a body or object
🏃♂ Motion can involve
Movement of body parts
Movement of the whole body
Or both together
➡ Types of motion
➖ Linear – movement in a straight line
e.g. ice hockey puck sliding
➰ Curvilinear – movement in a curved path
e.g. shot-put travelling through the air
🔄 Angular (rotational) – movement around an axis
e.g. gymnast rotating on a high bar
🔀 General motion – combination of linear and angular motion
🧠 Key idea for sport & exercise
General motion is most common in human movement
Even straight-line movement (e.g. running) requires limb rotation at synovial joints
📏 Vector and Scalar Measurements
➡ Vector
Has size and direction
Direction matters when combining measurements
Can only be directly combined if directions are the same
🔢 Scalar
Has size only
Can be added, subtracted, multiplied and divided easily
🧠 Why this matters in biomechanics
Affects how measurements are calculated and combined
Direction must be considered with vectors, but not with scalars
📍 Position
📌 Position
Described using coordinates
Measures distance from an origin
🗺 Coordinate systems
2D coordinates:
x = horizontal
y = vertical
3D coordinates:
x, y, z used for horizontal, vertical and lateral positions
Two common systems:
x = horizontal, y = vertical, z = lateral
x = horizontal, y = lateral, z = vertical
🔄 Angular position
Described using angles
Measured around one or more axes
➡ Linear kinematics
Focuses on motion in a straight line
📐 Linear Displacement vs Distance
🧭 Linear displacement (s)
Change in position from start to end
Includes how far and in which direction
Can be described horizontally, vertically and laterally
Vector quantity (size + direction)
SI unit: metres (m)
Direction given in degrees (°), radians (rad), or along an axis
📏 Linear distance (d)
The total path length travelled
Direction does not matter
Scalar quantity (size only)
🧠 Key distinction
Displacement = where you end up
Distance = how far you travel overall
➡ Linear velocity (v or u)
Rate of change in displacement over time
Includes speed and direction
Vector quantity
📐 Formula
𝑣= ∆𝑠/∆𝑡
Δs = change in displacement
Δt = change in time
Δ = “difference in”
⚡ Speed
The size (magnitude) of velocity
It is a Scalar quantity (no direction)
📏 Units
SI unit for velocity and speed: metres per second (m/s) or or m s-1
🧭 Direction
Can be given in degrees (°), radians (rad), or along a coordinate axis
Horizontal
Vertical
Lateral
Example: I run a 100m race in 20 seconds. Best time! I had an average velocity of 100m/20s = 5 m/s. This means that I moved about 5 meters every 1 second.
⚡ Linear Acceleration
Change in velocity over time
Can be linear or angular
Has size and direction → vector quantity
🔄 What counts as acceleration
Change in speed
Change in direction
Or both
Example: running around a bend at constant speed = accelerating
📐 Formula
𝑎= ∆𝑣/∆𝑡= ((𝑣−𝑢))/𝑡 where v is final velocity and u is initial velocity
v = final velocity
u = initial velocity
t = time taken for the change
📏 Units
SI unit: metres per second squared (m/s²or m/s/s or m s-2 )
🧭 Direction
Given in degrees (°) or radians (rad)
Or along a coordinate axis:
Horizontal
Vertical
Lateral
Example: The beginning of my 100m race I started from stop, and reached a maximum velocity of 6 m/s in 3 seconds. During this time I accelerated at 2m/s/s. Meaning that every second, my velocity increased by 2m/s.
Angular kinematics:
🌀 What it is
Study of rotational (spinning) motion around an axis
Applies to many sports movements
🏅 Sport applications
⛸ Figure skating: body rotation during jumps and spins
⚽ Soccer: leg rotating about the hip when kicking; ball spin affecting flight
🏊 Swimming: body rotation around the frontal axis during a flip turn
⚾ Baseball: ball spin during pitching; rotation of the arm and timing of swings
🏏 Cricket: bat rotation during a swing
📈 Performance links
Rotation and spin affect speed, trajectory, and movement outcome
Equipment length (e.g. cricket bat) can:
Increase angular velocity
Also increase moment of inertia, making movement harder
🎯 Key idea
Understanding angular kinematics helps improve performance and reduce injury risk
Angular displacement:
📐 Definition
Angular displacement is the angle by which an object rotates around a fixed point or axis. It measures the change in position of a rotating object and is expressed in radians (rad) or degrees (°)
🧭 Key features
Vector quantity → has size and direction
Symbol: θ (theta)
Direction: clockwise or anticlockwise
📏 Units
Measured in Degrees (°) or radians (rad)
Example 1: In gymnastics, when an athlete performs a somersault, their body rotates around an axis (usually their waist). If they complete a full flip, their angular displacement is 360° (or 2𝜋 radians) from their starting position.
Example 2: Flexing the elbow would yield a displacement of ~150° (depending on ROM)
Angular velocity:
⏱ Definition
Rate of change of angular displacement over time (is the rate at which an object rotates or spins around an axis. It measures how fast an object moves through an angle over time and is expressed in radians per second (rad/s))
🧭 Key features
Vector quantity → size and direction
Symbol: ω (omega)
Direction: clockwise or anticlockwise
📏 Units
Measured in Degrees per second (°/s or ° s-1); OR
Radians per second (rad/s or rad s-1)
🧮 Relationships
𝝎=𝜽/𝒕 but also, v = ωr
ω = angular velocity (rad/s or rad s-1)
r = radius of circle from the axis
𝒗=𝟐𝝅𝒓/𝑻
One full rotation = 2π radians
T = time for one full rotation
Angular acceleration:
⏱ Definition
Change in angular velocity divided by the time taken ((α) – is the rate at which an object's angular velocity (𝜔) changes over time. It describes how quickly an object speeds up or slows down its rotation and is measured in radians per second squared ("rad/s/s " or 〖𝑟𝑎𝑑 𝑠〗^(−2)))
🧭 Key features
Vector quantity → has size and direction
Symbol: α (alpha)
Direction: clockwise or anticlockwise about an axis
📏 Units
Degrees per second squared (°/s²)
Radians per second squared (rad/s²)
📐 Time and measurement
Velocity and acceleration depend on the time period measured
Measurements can be over long durations (e.g. whole race) or very short instants
As time → 0, motion is analysed using calculus
⚡ Instantaneous velocity & acceleration
Measured over a very short (“instant”) time
Can be found by:
📊 Using equations (via calculus)
📈 Using graphs:
Velocity = gradient of displacement–time graph
Acceleration = gradient of velocity–time graph
📊 Average velocity & acceleration
Calculated over a longer time period
Simpler to calculate than instantaneous values
➗ Formulas
🏃 Average velocity = change in displacement ÷ time taken
🚀 Average acceleration = change in velocity ÷ time taken
Kinematics: Instantaneous vs average:
Instantaneous velocity and acceleration refers to the measurements at any one point in time
Average refers to the overall measurement
Ex. If I run a 100m dash, I start from stop. This means that I will accelerate for a few seconds to my maximum velocity. If it takes me 20 seconds to run this, I had an average velocity of 5m/s, but because I accelerated at the start, there were periods in the beginning of my race that had instantaneous velocities lower than 5m/s, and for quite a lot of the race I had instantaneous velocities greater than 5m/s
Additionally, during the first 3 seconds, I had instantaneous acceleration of 2m s-2, but when averaged with the end of the race (where I slowed down) the average accelerations is only 0.5 m s-2
🔍 What is kinetics?
Study of the forces involved in movement
Can be:
➡ Linear kinetics (straight-line motion)
🔄 Angular kinetics (motion around an axis)
💥 Force
🤝 Definition
A mechanical interaction between two objects or bodies
Can act:
✋ With contact (e.g. friction)
🌍 At a distance (e.g. gravity)
➡ Effect of force
Forces change or attempt to change motion
Resultant motion depends on the sum of all forces acting
In sport, movement is determined by the magnitude and direction of forces
🤸 Sport example
Trampolinist’s motion depends on:
⬆ Force from the trampoline
⬇ Force of gravity
Adjusting forces can increase:
Jump height
Time in the air
Opportunity for skills (e.g. twists, somersaults)
🌍 Gravity
📜 Newton’s contribution (1687)
Gravity is an attractive force between objects with mass
Strength depends on:
Mass of objects
Distance between them
🌌 Effects of gravity
Causes planetary orbits
Attracts objects toward the centre of the Earth
Acts vertically on bodies near Earth’s surface
⚖ Mass and Weight
🧱 Mass
Amount of material in an object
Measured in kilograms (kg)
Remains constant regardless of location
⬇ Weight
Force caused by gravity acting on mass
Depends on the strength of gravity
Example:
Same mass on Earth and Moon
Lower weight on the Moon due to weaker gravity
Newton's laws:
📘 Context
Newton’s three laws explain how forces relate to motion
They accurately describe motion in everyday objects and the human body
Less accurate at:
🚀 Speeds near the speed of light
⚛ Subatomic scales
First Law – the law of inertia. An object at rest stays at rest, or an object in motion stays in motion UNLESS acted on by a force. Inertia is resistance to change in movement
Second Law – the law of acceleration. The acceleration of an object is proportional to the force acting on it, and inversely related to its mass.
𝐹=𝑚∙𝑎 ; 𝐹_𝑔=𝑚∙𝑔 ; 𝐹=(𝑚(𝑣_𝑓−𝑣_𝑖))/𝑡
Third Law – the law of reaction. For every action there is an equal an opposite reaction. Forces are the same, but results might not be the same.
Newton's first law:
📜 Statement
An object will remain at rest or continue with constant velocity
Unless acted on by an unbalanced force
🧠 Key ideas
Objects do not change motion without an external force
Also known as the law of inertia
⚖ Balanced forces
If an object is:
🧍 At rest → forces are balanced
➡ Moving at constant speed in a straight line → forces are balanced
⬆⬇ Reaction force
On Earth, weight (gravity) acts downward
An equal reaction force (from the ground/support) acts upward
🎯 Important clarification
Forces do not cause motion
Forces cause changes in motion (start, stop, speed up, slow down, change direction)
Newton's second law:
📜 Statement
Acceleration of an object is proportional to the unbalanced force
Acceleration is in the same direction as the applied force
Applies to objects with constant mass
🧮 Key equation
F = ma
💪 F = force
⚖ m = mass
🚀 a = acceleration
📈 Key relationships
🔼 Greater force → 🔼 greater acceleration
⚖ Greater mass → 🔽 less acceleration for the same force
🏋 Heavier objects require larger forces to accelerate
⏱ Using velocity
Acceleration = change in velocity ÷ time
F = m(v − u) / t
v = final velocity
u = initial velocity
t = time taken
🌍 Gravity and weight
Force due to gravity (weight): Fg = mg
m = mass
g = acceleration due to gravity
Newton's third law:
📜 Statement
When one object applies a force to another, the second applies a force equal in size and opposite in direction
🗣 Common phrasing
“For every action, there is an equal and opposite reaction”
🔍 Key points
👥 Forces act on two different objects, not the same one
⚖ Forces are equal in magnitude, regardless of object mass
📊 Effects can differ due to different masses (Newton’s 2nd law)
⏱ Forces occur simultaneously, not one after the other
🧍 Centre of mass
⬇ Lower centre of mass → greater stability
📐 Base of support
🔲 Larger base of support → greater stability
📍 Line of gravity
✔ Line of gravity within the base of support → greater stability
⚖ Mass
🧱 Greater mass → greater stability
💪 Muscle forces
Muscles generate forces that act on joints to produce movement (e.g. running, jumping, throwing)
🦵 Multiple forces at joints
Each joint experiences several forces at the same time
Example: during a volleyball jump, leg muscles act at the knee while feet push against the ground
➕ Forces working together
When muscle forces act in the same direction, they combine
Combined forces result in more powerful movements (e.g. jumping higher)
🎯 Performance implication
Effective summing of joint forces improves movement and sporting performance
⚠ Injury risk
Excessive force or poor technique can increase joint stress and raise injury risk
🏃 Linear momentum
Property an object has due to its movement
Calculated as mass × velocity
Vector quantity (has size and direction)
Measured in kg·m·s⁻¹
Represented by:
p = mv
⏱ Linear impulse
Force multiplied by the time the force acts
Vector quantity
Represented by:
J = FΔt
🔄 Impulse–momentum relationship
Linear impulse = change in linear momentum
Change in momentum depends on:
size of the force
time the force acts
Large force and/or long time → large change in momentum
As mass is usually constant, a change in momentum results in a change in velocity
🟢 Impulse = force applied over time
Acts like a “push” applied to an object (e.g. a ball)
➡ Direction of impulse determines direction of motion
The object moves in the same direction as the applied force
⚽ Application in sport
In ball sports (e.g. soccer, basketball), applying force in the correct direction controls where the ball travels
🎯 Key idea
Applying the right amount of force in the right direction allows athletes to control movement and outcomes (e.g. passes, shots, kicks)
🔧 Torque (moment of force)
Created when a force is applied to an object that can rotate about an axis
The force must not act directly through the axis
The applied force is called an eccentric force
📏 Factors affecting the size of torque
💪 Size of the force
➡ Direction of the force
📐 Distance from the axis of rotation
🦴 Importance in the human body
Most body segments rotate around axes at synovial joints
Muscle arrangement relative to joints determines how much torque is produced
🧱 Link to levers
Bones act as rigid rods
Joints act as axes of rotation
Principle of moment of inertia
⚙ Definition
How difficult it is for a body or object to rotate about an axis
⚖ Measurement
Measured in kg m²
📦 Factors affecting moment of inertia
Mass of the body or object
Distribution of mass relative to the axis
More mass further from the axis → greater moment of inertia (harder to rotate)
More mass closer to the axis → smaller moment of inertia (easier to rotate)
🧍 Human movement
The body’s moment of inertia changes with body position and axis of rotation
closely linked to lever systems
Principle of angular momentum
🔁 Definition
A measure of the amount (or potential) of rotation
🔧 How it is generated
Produced by torque from an eccentric force acting on a body free to rotate
Forces may come from the ground, equipment, another body, or muscles
📐 Properties
Vector quantity (has size and direction)
Measured in kg m s⁻¹
🧮 Formula
L = Iω
I = moment of inertia
ω = angular velocity
🦴 Human body applications
Generated by applying torque to the ground or apparatus
Reaction forces (Newton’s third law) create body rotation
Smaller moment of inertia than the Earth → greater rotation of the body
🤸 Sporting relevance
Essential for rotations in gymnastics (somersaults, twists)
Also important in golf swings, javelin throws and football kicks
Principle of conservation of angular momentum:
🌀 Angular form of Newton’s first law
A rotating body continues to rotate with constant angular momentum unless acted on by an external unbalanced torque
🔒 Key principle
Once angular momentum is generated, it remains constant unless an external torque changes it
🤸 Airborne human movement
In sports such as gymnastics, diving, long jump and high jump, the main forces during flight are gravity and air resistance
🌍 Effect of gravity in flight
Gravity acts through the centre of gravity
Axes of rotation during flight also pass through the centre of gravity
Therefore, gravity creates no torque (distance from axis = 0)
💨 Air resistance
Effect is very small due to low flight speeds and relatively large body mass
⚖ Result
With no external torque acting, angular momentum is conserved during flight
🧮 Angular momentum relationship
Angular momentum = moment of inertia × angular velocity
🔄 Changes during flight
↓ Moment of inertia → ↑ Angular velocity
↑ Moment of inertia → ↓ Angular velocity
🤸♂ Sport example
Tucking in a somersault reduces moment of inertia → faster rotation
Opening out before landing increases moment of inertia → slower rotation
❌ Common misconception
Changes in rotation speed are not caused by air resistance
Transfer of angular momentum:
🚫 No creation or destruction in flight
Once airborne, total body angular momentum cannot change (no external torque)
⚖ Segment interaction
If one body part speeds up, another must slow down
This keeps total angular momentum constant
🔁 Continuous transfer
When the first part later slows, the second part speeds up again
🏊 Piked dive example
During hip flexion (0–0.5 s) into a piked position
⬆ Upper body angular velocity increases
⬇ Lower body angular momentum decreases
🎯 Key principle
Greater angular momentum in one segment = less in another
Total body angular momentum is conserved
Trading angular momentum:
🧭 Angular momentum is a vector
It has size and direction
🤸 Multiple axes of rotation
An athlete may rotate about one axis (e.g. somersaulting)
🔀 Introducing a second rotation
Adding rotation about another axis (e.g. tilting) using body segments
🌀 Resulting motion
The combination of two angular momentum vectors creates rotation about a third axis (twisting)
🎯 Application in gymnastics
Used to create twisting during somersaults
♻ Key idea
Some somersault angular momentum is traded for twisting angular momentum
💥 Collision
A collision is the physical contact of two or more objects for a short time.
⚖ Conservation of Linear Momentum
Momentum = mass × velocity.
When a collision happens, the overall momentum remains the same.
🏃♂➡🏃♀ Example
If a 100 kg person moving at 10 m/s hits a 50 kg person who is stationary, and all momentum is transferred, the larger person stops and the smaller person moves away at 20 m/s.
🔥🔊 Energy Loss
Momentum is conserved, but some energy may be lost to the surroundings as heat and sound, resulting in lower final speeds.
📊 Coefficient of Restitution (Cᵣ)
For two objects a and b:
𝐶_𝑟= (𝑉_𝑓𝑏 − 𝑉_𝑓𝑎)/(𝑉_𝑖𝑎−𝑉_𝑖𝑏 )
🎯 Interpretation
Cᵣ close to 1 → elastic collision (little or no energy lost)
🧲 Friction
Friction is a force that acts against motion when two surfaces are in contact
The frictional force acts parallel to the surface of contact
👟🌍 Example: Walking or Running
Shoes push against the ground
Friction acts in the opposite direction to the applied force, helping movement
⬇ Body Weight
Force of gravity acting downward on the body
⬆ Normal Force
Force exerted by the ground on the body, acting upward
📌 Types of Friction
Static (limiting) friction
Dynamic friction
⛔ Static friction acts between two surfaces not moving relative to each other
🧱 It prevents motion from starting when a force is applied
🔢 The coefficient of static friction (μₛ) describes how much friction exists between two surfaces
🧊 Smooth surfaces (e.g. steel on ice) → low μₛ
👟 Rough surfaces (e.g. rubber on ground) → high μₛ
📊 μₛ is a scalar quantity and usually ranges between 0 and 1 (sometimes higher)
⚖ Formula:
𝜇_𝑠=𝐹_𝑓/𝐹_𝑁 or 𝐹_𝑓≤ 𝜇_𝑠∙ 𝐹_𝑁
(where Fᶠ = friction force, Fᴺ = normal force)
📈 Friction force increases with applied force up to a maximum:
Fᶠ ≤ μₛFᴺ
🚦 Maximum static friction occurs just before motion begins
▶ Dynamic friction occurs once an object is moving
🚦 Motion begins when the applied force overcomes static friction
📉 The friction force decreases once motion starts (lower than static friction)
🔄 Dynamic friction acts between surfaces moving relative to each other
🔢 The coefficient of dynamic friction (μᵈ) describes friction during motion
⚖ Formula:
μᵈ = Fᶠ / Fᴺ
(where Fᶠ = friction force, Fᴺ = normal reaction force)
🧮 Rearranged form:
Fᶠ = μᵈFᴺ
Friction in sport:
⚖ Friction can be increased or decreased to improve performance, depending on the sport
🏸 High friction for control and speed changes
Enables quick starts, stops, and direction changes
Example: Badminton shoes designed with a high coefficient of friction
🤸 Increased grip for safety and force application
Chalk used by gymnasts on bars and rings
Increases friction to prevent slipping and allow greater force application
🎿 Low friction for speed
Athletes aim to reduce friction to move faster
Example: Downhill skiers wax skis to decrease friction
Work and power:
🧮 Work
➡ Work is force applied over a distance
➡ 𝑊 = 𝐹 · 𝑑
➡ 🧪 W = work (joules)
➡ 💪 F = force (newtons)
➡ 📏 d = distance (metres)
🔄 Energy transfer
➡ Work transforms energy from one form to another
⚡ Power
➡ Power is the rate at which work is done
➡ 𝑃= ∆𝑊/∆𝑡,
➡ ⚡ P = power (watts)
➡ ⏱ Δt = time
➡ Alternative form: 𝑃 = 𝐹 · 𝑣 (force × velocity)
🔥 High power output
➡ Large power output = lots of work done quickly
➡ In exercise, this relates to high intensity
🏋 Optimising power in sport
➡ Muscular power = strength + speed
➡ High force with slow movement = low power
🎯 Technique
➡ Correct technique allows greater forces to be produced
🛠 Equipment
➡ Equipment can increase force or capture more work
➡ Example: 🚴 Clip-in cycling shoes allow both pushing and pulling during pedalling
Power output in sport:
➡ Muscular power = strength + speed of movement
➡ Power is a key component of successful performance in many sports
➡ Optimising power improves functional sport performance
🎯 Optimising Power Through Technique
➡ Correct technique maximises force production
➡ Correct technique maximises speed of movement
➡ Greater force applied quickly = higher power output
🦵 Example: Trampolining
➡ Bending knees increases force applied
➡ Faster knee extension increases movement speed
➡ Combining high force + high speed results in greater height
🚴 Optimising Power Through Equipment
➡ Equipment design affects efficiency and power output
➡ Effective equipment allows greater force application
➡ Example: clip-in cycling shoes increase average and maximum sprint power
🏆 Key Idea
➡ Best performance comes from combining correct technique and well-designed equipment