Leaving Certificate Physics
⚙️ Mechanics
Your complete guide to motion, forces, energy, pressure, gravity, circular motion and more — explained simply with worked examples.
📘 Chapter I: Motion & Forces
📗 Chapter II: Energy, Pressure & Gravity
📙 Chapter III: Vectors, Circular Motion & SHM
Speed & Velocity
Displacement, speed, distance-time graphs
Chapter I
Equations of Motion
UVAST method, 3 key equations
Chapter I
Newton's Laws
Force, mass, F=ma, free body diagrams
Chapter I
Momentum
Conservation of momentum, collisions
Chapter I
Work, Energy & Power
KE, PE, efficiency, solar constant
Chapter II
Density & Pressure
Pascal, Archimedes, Boyle's Law
Chapter II
Gravity
Newton's law of gravitation, g=GM/d²
Chapter II
Moments & Equilibrium
Turning effect, levers, torque
Chapter II
Vectors
Resolving, inclined planes, projectiles
Chapter III
Circular Motion
Angular velocity, centripetal force
Chapter III
SHM
Simple harmonic motion, pendulums
Chapter III
All Formulas
Quick reference sheet
Revision
💡 How to use this guide: Work through each section in order using the sidebar. Every section has simple explanations, key definitions to learn, formulas, and worked examples. Quizzes test you at the end of each booklet!
Chapter I — Motion
📏 Speed, Velocity & Displacement
The difference between distance and displacement, and between speed and velocity, is one of the most important ideas in all of mechanics.
📐 Displacement vs Distance
📖 Definition — Displacement
Displacement is the straight-line distance between two points, in a given direction.
Distance is how far you actually travel along a path. Displacement is "as the crow flies" — the straight-line measurement from start to finish, with a direction.
🧠 Example: If you run halfway around a circular track of radius 100 m, the distance you travelled is 100π ≈ 314 m. But your displacement is only 200 m (the diameter) — the straight line from start to finish.
⚡ Speed vs Velocity
📖 Definition — Speed
Speed is the rate of change of distance with respect to time. (Speed = distance ÷ time)
📖 Definition — Velocity
Velocity is the rate of change of displacement with respect to time. (Velocity = displacement ÷ time)
Speed
scalar
Just a number — no direction. E.g. "60 km/h"
Velocity
vector
Number AND direction. E.g. "60 km/h north"
Unit of both
m s⁻¹
Metres per second (m/s)
🧠 Key trick: If you run a full lap of a circular track and finish where you started, your average velocity is ZERO (displacement = 0). But your average speed is definitely not zero!
📊 Distance-Time Graphs
On a distance-time graph, the slope (gradient) of the graph = speed. A steeper slope = faster speed. A flat (horizontal) line = object is stopped.
⚡ Instantaneous vs Average Speed
Average speed = total distance ÷ total time (for the whole journey). Instantaneous speed = speed at one exact moment (what a speedometer shows).
🕰️ Ticker Tape Timer
A ticker tape timer makes 50 dots per second on a paper tape. The time between adjacent dots = 1/50 = 0.02 seconds.
- To measure speed: measure the distance between 11 dots (= 10 intervals), divide by (10 × 0.02) = 0.2 s
- If dots are equally spaced → constant speed. If getting farther apart → accelerating.
🎯 Exam tip: Always use 10 intervals (11 dots) rather than 2 dots to reduce percentage error — a single measurement is less reliable than an average over 10 intervals.
Chapter I — Motion
📈 Acceleration & Velocity-Time Graphs
Acceleration is about changing velocity. Velocity-time graphs are one of the most important tools in mechanics.
📖 Definition — Acceleration
Acceleration is the rate of change of velocity with respect to time.
Formula — Acceleration
a = (v − u) / t
a = acceleration (m s⁻²) v = final velocity (m s⁻¹) u = initial velocity (m s⁻¹) t = time (s)
The unit of acceleration is the metre per second squared (m s⁻²).
📊 Velocity-Time (v-t) Graphs
On a velocity-time graph:
- Slope (gradient) = acceleration
- Area under the graph = distance travelled
- Flat horizontal line = constant velocity (zero acceleration)
- Sloping up = accelerating Sloping down = decelerating
🧠 Finding the area: Break the graph into triangles and rectangles. Area of triangle = ½ × base × height. Area of rectangle = base × height.
🧪 Measuring Acceleration — Ticker Tape
To measure acceleration using ticker tape:
- Measure 5 intervals near the start of tape → calculate initial velocity u = s₁/t₁
- Measure 5 intervals near the end of tape → calculate final velocity v = s₂/t₂
- Measure time t from the centre of u to the centre of v
- Calculate a = (v − u) / t
🎯 Experiment: Measuring Acceleration due to Gravity (g)
Apparatus: Millisecond timer, metal ball bearing, trapdoor, electromagnet.
Method: Release ball → timer starts → ball hits trapdoor → timer stops. Repeat for different heights s. Plot graph of s vs t². Slope = g/2, so g = 2 × slope.
Precaution: Use paper between ball and electromagnet to ensure clean release. Take smallest t for each s (excess time = residual magnetism in electromagnet).
Apparatus: Millisecond timer, metal ball bearing, trapdoor, electromagnet.
Method: Release ball → timer starts → ball hits trapdoor → timer stops. Repeat for different heights s. Plot graph of s vs t². Slope = g/2, so g = 2 × slope.
Precaution: Use paper between ball and electromagnet to ensure clean release. Take smallest t for each s (excess time = residual magnetism in electromagnet).
Chapter I — Motion
🧮 Equations of Motion — UVAST
Three equations that link five quantities. Learn the UVAST method and you can solve any motion problem.
The 5 UVAST Variables
U
Initial velocity (speed at the START)
m s⁻¹
V
Final velocity (speed at the END)
m s⁻¹
A
Acceleration
m s⁻²
S
Displacement (distance travelled)
m
T
Time
s
The Three Equations
⚠️ Equation 1
v = u + at
Use when you know/want: v, u, a, t (not s)
⚠️ Equation 2
s = ut + ½at²
Use when you know/want: s, u, a, t (not v)
⚠️ Equation 3
v² = u² + 2as
Use when you know/want: v, u, a, s (not t)
Bonus Formula
s = (u + v)/2 × t
Also useful — average velocity × time
🪄 The UVAST Method
- Write U, V, A, S, T down the left-hand side of your page
- Fill in the values you know. Put ? beside what you want to find
- Write out all 3 equations
- Choose the equation that has exactly ONE unknown
- Substitute and solve
🧠 Magic rule: You need exactly 3 known values to find the 4th and 5th. Once you know 3, you can work out everything!
⚠️ Signs matter! Pick a positive direction (e.g. upward = positive). Anything going the opposite way gets a minus sign. For objects thrown upward, g = −9.8 m s⁻².
🍎 At the highest point: v = 0 (velocity is zero at the peak). The object is still accelerating downward at 9.8 m s⁻² even at the top!
Chapter I — Motion
🍎 Gravity & Free Fall
On Earth, every falling object accelerates downward at the same rate — regardless of mass!
📖 Definition — Acceleration due to Gravity (g)
The acceleration due to gravity is the acceleration experienced by all objects in free fall near the surface of the Earth, caused by the Earth's gravitational attraction.
Value of g on Earth
g = 9.8 m s⁻²
Always acting downward
📌 Key Rules for Gravity Problems
- If object is released from rest: u = 0
- At the highest point: v = 0
- If something moves UP, take up as positive → a = −9.8 m s⁻²
- Objects of different mass fall at THE SAME rate (in the absence of air resistance)
- A stone thrown upward is still accelerating downward even while it's moving up!
🧠 Why does direction of acceleration confuse people? The direction of acceleration is determined by the direction of the FORCE causing it — not the direction of motion. Gravity always acts down, so the acceleration is always downward — even when the object is moving upward!
🌕 Moon vs Earth
On the Moon, g ≈ 1.6 m s⁻². Objects still fall — but much more slowly. You could throw a stone 6× higher on the Moon with the same initial speed!
Chapter I — Motion
💪 Force & Newton's Laws
Newton figured out three simple laws that describe how forces and motion are related. They're still used by NASA today!
📖 Definition — Force
A force is anything which has the potential to cause an object to accelerate. Unit: newton (N).
⚖️ Newton's Three Laws
📖 Newton's First Law
Every object will continue to travel at constant velocity unless an unbalanced external force acts on it.
🧠 Simple version: Objects keep doing what they're doing unless a force changes it. If it's moving, it keeps moving. If it's still, it stays still. Inertia!
📖 Newton's Second Law
The rate of change of an object's momentum is directly proportional to the force which caused it, and takes place in the direction of the force.
⚠️ F = ma (special case of 2nd Law)
F = ma
F = net force (N) m = mass (kg) a = acceleration (m s⁻²)
⚠️ Always remember: AN UNBALANCED FORCE PRODUCES AN ACCELERATION
📖 Newton's Third Law
When objects interact they exert equal but opposite forces on each other.
🚀 Rocket example: The rocket pushes gas downward (action). The gas pushes the rocket upward (reaction). Equal and opposite forces!
🧮 Net Force (Resultant Force)
The net force is the sum of all forces acting. Forces in the same direction add together. Forces in opposite directions subtract.
Example
Forces of 8 N right and 6 N left → Net force = 8 − 6 = 2 N to the right
🪨 Friction
📖 Friction
Friction is a force which opposes the relative motion between two objects.
Friction is calculated using: Net force = Engine force − Friction force. If an object moves at CONSTANT velocity, the net force = 0, so friction = engine force.
🌊 Terminal Velocity
A skydiver accelerates downward because gravity > air resistance. As speed increases, air resistance grows. Eventually air resistance = gravity → net force = 0 → constant speed = terminal velocity (≈ 100 m/s).
⚖️ Mass vs Weight
Mass
kg
How much matter. Does NOT change with location. Measured in kg.
Weight
N
The gravitational force on the object. W = mg. Changes on the Moon!
Weight
W = mg
W = weight (N) m = mass (kg) g = 9.8 N kg⁻¹ on Earth
Chapter I — Motion
🎯 Momentum & Collisions
Momentum is like "quantity of motion". It's a vector, and in any collision it's always conserved!
📖 Definition — Momentum
Momentum (ρ) = mass × velocity. Unit: kg m s⁻¹ or N s.
Formula
ρ = mv
ρ = momentum (kg m s⁻¹) m = mass (kg) v = velocity (m s⁻¹)
🔁 Principle of Conservation of Momentum
📖 Definition
In any interaction, the total momentum before the interaction equals the total momentum afterwards, provided no external forces act on the system.
⚠️ Don't forget: The phrase "provided no external forces act on the system" is worth marks in the exam — don't leave it out!
Three Types of Collision
Hit 'n Bounce
Objects collide and separate.
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Hit 'n Stick
Objects stick together after collision (coalesce). One combined velocity v₃.
m₁u₁ + m₂u₂ = (m₁+m₂)v₃
m₁u₁ + m₂u₂ = (m₁+m₂)v₃
Explosion
Objects start at rest (total p = 0). They fly apart with equal and opposite momentum.
0 = m₁v₁ + m₂v₂
0 = m₁v₁ + m₂v₂
⚠️ Directions matter! Momentum is a vector. If two objects move in opposite directions, one gets a negative sign. This is how the gun recoil and explosion type questions work — the total momentum is still zero!
💡 Kinetic energy in collisions: Momentum IS conserved in collisions. Kinetic energy is NOT (some is converted to heat and sound). This is important for exam questions!
🔬 Experiment: Verifying Conservation of Momentum
Two trolleys on a track — Trolley A (moving) collides with Trolley B (stationary). They coalesce via velcro. Use ticker tape to measure velocity before and after. Show that m₁u₁ ≈ (m₁+m₂)v₃.
🎯 Precautions: Tilt the track slightly to compensate for friction so trolleys move at constant velocity. Ensure the track is smooth and doesn't sag.
Chapter I — Motion
⚡ Impulse & the Newton
Impulse explains why airbags save lives — and why cricketers pull their hands back when catching.
📖 Definition — Impulse
Impulse is the change in momentum. Impulse = Ft = mv − mu
⚠️ Impulse Formula
Ft = mv − mu
F = force (N) t = time of contact (s)
mv − mu = change in momentum (kg m s⁻¹)
mv − mu = change in momentum (kg m s⁻¹)
🛡️ How airbags save lives: From the formula Ft = mv − mu, the change in momentum (right-hand side) is FIXED (it depends on the crash). But Ft on the left can vary — if the time t increases (airbag slows the head down gradually), then the force F must decrease. Less force = less injury. This is also why there are crumple zones in cars!
🏏 Cricket / Golf Examples
A cricket player pulls their hands back when catching the ball. This increases the contact time t, which from Ft = mv − mu means the force F on their hands is smaller → less pain, easier to hold on!
A golfer follows through with their swing → increases contact time → more impulse → ball goes faster.
📖 The Newton (unit of force)
One newton is the force which causes a body of mass 1 kg to undergo an acceleration of 1 m s⁻².
🧠 Newton's Laws and Concussion
When a rugby player gets tackled: skull stops suddenly (Newton's 1st Law) but brain keeps moving forward until it hits the skull. Force = rate of change of momentum (Newton's 2nd Law). The skull exerts an equal force back on the brain (Newton's 3rd Law). A helmet increases the impact time → reduces force → reduces concussion risk.
Chapter I — Worked Examples
📐 Calculations: Chapter I
Step-by-step worked examples from your notes. Pay attention to the UVAST method!
Example 1 — Basic Acceleration
Q: A car accelerates from 10 m s⁻¹ to 30 m s⁻¹ in 5 s. Find the acceleration.
Write UVAST
U = 10 m s⁻¹, V = 30 m s⁻¹, A = ?, S = —, T = 5 s
Use v = u + at
30 = 10 + a(5)
a = (30−10)/5 = 4 m s⁻²
a = (30−10)/5 = 4 m s⁻²
✅ Acceleration = 4 m s⁻²
Example 2 — Distance from Rest
Q: A car starts from rest with acceleration 6 m s⁻². Find the distance in 12 s.
UVAST: U=0, V=?, A=6, S=?, T=12
Use s = ut + ½at²
Substitute
s = (0)(12) + ½(6)(12²)
s = 0 + 3 × 144 = 432 m
s = 0 + 3 × 144 = 432 m
✅ Distance = 432 m
Example 3 — Object Thrown Upward
Q: A stone is thrown upward at 20 m s⁻¹. How high after 2 s?
UVAST: U=20, V=?, A=−9.8, S=?, T=2
Taking up as positive, a = −9.8 m s⁻²
s = ut + ½at²
s = (20)(2) + ½(−9.8)(2²)
s = 40 − 19.6 = 20.4 m
s = 40 − 19.6 = 20.4 m
✅ Height after 2 s = 20.4 m
Example 4 — Newton's Laws
Q: A car (750 kg) accelerates at 1.2 m s⁻². Engine force = 2000 N. Find: (i) net force, (ii) friction force.
Part (i) — Net force: F = ma
F_net = (750)(1.2) = 900 N
Part (ii) — Friction: F_net = F_engine − F_friction
900 = 2000 − F_friction
F_friction = 2000 − 900 = 1100 N
F_friction = 2000 − 900 = 1100 N
✅ Net force = 900 N, Friction = 1100 N
Example 5 — Hit 'n Stick Collision
Q: Trolley A (12 kg, 3.5 m s⁻¹) hits stationary Trolley B (12 kg). They stick. Find common velocity.
Conservation of momentum: m₁u₁ + m₂u₂ = (m₁+m₂)v₃
(12)(3.5) + (12)(0) = (12+12)(v₃)
42 = 24v₃
42 = 24v₃
Solve for v₃
v₃ = 42/24 = 1.75 m s⁻¹
✅ Common velocity = 1.75 m s⁻¹
Example 6 — Impulse (Golf Ball)
Q: Force of 5300 N on a golf ball (45 g) for 0.54 ms. Find speed of ball.
Use Ft = mv − mu. Ball starts at rest (u=0)
F = 5300 N, t = 0.54 × 10⁻³ s, m = 0.045 kg
Substitute
(5300)(0.54 × 10⁻³) = (0.045)(v) − 0
2.862 = 0.045v
v = 63.6 m s⁻¹
2.862 = 0.045v
v = 63.6 m s⁻¹
✅ Speed = 63.6 m s⁻¹
Chapter I — Quiz
✅ Quiz: Chapter I
Test yourself on Motion, Forces and Momentum!
Chapter II — Energy & Power
⚡ Work, Energy & Power
Three closely related concepts that explain how forces transfer energy and how fast machines work.
🔧 Work
📖 Definition — Work
Work is defined as the product of force and displacement. W = F × s. Unit: joule (J).
Formula — Work
W = F × s
W = work done (J) F = force (N) s = displacement (m)
⚡ Energy
📖 Definition — Energy
Energy is the ability to do work. Unit: joule (J).
Kinetic Energy (energy of motion)
EK = ½mv²
EK = kinetic energy (J) m = mass (kg) v = velocity (m s⁻¹)
Potential Energy (energy of position)
EP = mgh
EP = potential energy (J) m = mass (kg) g = 9.8 N kg⁻¹ h = height (m)
🔌 Power
📖 Definition — Power
Power is the rate at which work is done. Unit: watt (W). 1 W = 1 J s⁻¹.
Formula — Power
P = W/t = E/t = Fv
P = power (W) W = work done (J) t = time (s)
💡 Percentage Efficiency
Efficiency
% Efficiency = (useful power out / power in) × 100
No machine is 100% efficient — some energy is always lost as heat.
☀️ The Solar Constant
📖 The Solar Constant
The solar constant is the energy from the Sun falling on each square metre of the Earth's surface per second. Value ≈ 1360 W m⁻².
🧮 How to use it: Total energy = solar constant × area × time (in seconds). E.g. if solar constant = 1360 W m⁻², area = 5 m², time = 60 s: Energy = 1360 × 5 × 60 = 408,000 J.
🔌 The Kilowatt-Hour (kWh)
Used on electricity bills. 1 kWh = energy used by a 1000 W appliance in 1 hour.
kWh Calculation
No. of kWh = kilowatts × hours
Cost = kWh × price per unit
Chapter II — Energy
🔋 Principle of Conservation of Energy
Energy can never be created or destroyed — it just changes form!
📖 Principle of Conservation of Energy
Energy cannot be created or destroyed — it can only be converted from one form to another.
🎢 Roller coaster example: At the top (low speed, high height) → lots of PE, little KE. At the bottom (high speed, low height) → lots of KE, little PE. Total energy stays the same (ignoring friction).
📐 Key Energy Equation
Loss in PE = Gain in KE (for freely falling object)
mgh = ½mv²
Can divide both sides by m: gh = ½v² So: v = √(2gh)
🎯 Work-Energy Theorem: The work done on an object equals the change in its energy. Fs = ½mv² (if starting from rest). This shows that stopping distance is proportional to v² — doubling speed means 4× the stopping distance!
🏀 Bouncing Ball Example
A ball at height 3 m has PE = mgh. When it hits the ground, all PE → KE. If it loses 6 J on bounce, the remaining KE becomes PE for the next bounce. New height = remaining KE / (mg).
Chapter II — Density & Pressure
🪣 Density & Pressure
Why does a boat float? Why does a thumb tack work? It's all about density and pressure!
🧊 Density
📖 Definition — Density
Density is defined as mass per unit volume. ρ = m/V. Unit: kg m⁻³.
Formula — Density
ρ = m / V
ρ (rho) = density (kg m⁻³) m = mass (kg) V = volume (m³)
⚠️ Unit conversion: 1 m³ = 1,000,000 cm³ = 10⁶ cm³. Be careful when converting between cm³ and m³!
🔵 Pressure
📖 Definition — Pressure
Pressure is defined as force per unit area. P = F/A. Unit: pascal (Pa). Pressure is a SCALAR quantity (no direction).
Formula — Pressure
P = F / A
P = pressure (Pa) F = force (N) A = area (m²)
🧠 Why does a thumb tack work? The force is the same whether you push on the flat head or the point. But the area at the point is tiny, so the pressure there is enormous — enough to penetrate wood!
💧 Pressure in a Fluid
Pressure in a Fluid
ΔP = ρgΔh
ΔP = change in pressure (Pa) ρ = density (kg m⁻³) g = 9.8 m s⁻² Δh = depth (m)
Pressure in a fluid increases with depth. This is why your ears hurt when you dive deep underwater!
Chapter II — Pressure
🌊 Archimedes' Principle & Boyle's Law
Why do ships float? Why does a balloon shrink when you squeeze it? Archimedes and Boyle have the answers.
📖 Archimedes' Principle
When an object is immersed in a fluid, the upthrust it experiences is equal to the weight of the displaced fluid.
📖 Law of Flotation
A floating object displaces a weight of fluid equal to its own weight.
🧠 Simple version: An object floats if it displaces fluid that weighs MORE than the object itself. A ship is made of steel (dense) but it's mostly hollow air, so the average density of the ship is less than water → it floats!
🫧 Boyle's Law
📖 Boyle's Law
At constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure.
⚠️ Boyle's Law Formula
P₁V₁ = P₂V₂
Pressure 1 × Volume 1 = Pressure 2 × Volume 2 (at constant temperature)
🧠 Simple explanation: Squeeze a balloon — volume goes down, pressure goes up. The product PV stays the same! It's like a seesaw — when P goes up, V goes down proportionally.
🧪 Experiment: Verifying Boyle's Law
Use Boyle's Law apparatus — compress air with a screw. Record pressure and volume at each step (7+ readings). Plot P against 1/V → straight line through origin → verifies that P ∝ 1/V.
🎯 Why wait between adjustments? Changing pressure quickly generates heat which changes the temperature. Wait for gas to return to room temperature before recording. Boyle's Law only applies at CONSTANT temperature!
Chapter II — Gravity
🌍 Newton's Law of Gravitation
Every object in the universe attracts every other object. Newton figured out the exact mathematical rule.
📖 Newton's Law of Universal Gravitation
Any two point masses in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
⚠️ Newton's Law of Gravitation
F = Gm₁m₂ / d²
F = gravitational force (N)
G = universal gravitational constant = 6.7 × 10⁻¹¹ N m² kg⁻² (in your tables)
m₁, m₂ = the two masses (kg)
d = distance between centres of the masses (m)
G = universal gravitational constant = 6.7 × 10⁻¹¹ N m² kg⁻² (in your tables)
m₁, m₂ = the two masses (kg)
d = distance between centres of the masses (m)
🧮 Gravitational Field Strength at Any Point
⚠️ Gravitational Field Strength
g = GM / d²
g = gravitational field strength at distance d from centre of planet (N kg⁻¹)
M = mass of the planet (kg) d = distance from centre (m)
M = mass of the planet (kg) d = distance from centre (m)
🌙 Key facts: Mass stays the same wherever you go. Weight changes because g changes. On the Moon, g ≈ 1.6 m s⁻², so you weigh about 1/6 of your Earth weight — but your mass is still the same!
⚖️ Mass vs Weight — Summary
| Property | Mass | Weight |
|---|---|---|
| What is it? | Amount of matter (inertia) | Gravitational force on the object |
| Unit | kg | N (newton) |
| Changes on Moon? | No — stays the same | Yes — much less (W = mg) |
| Measured by | Balance scales | Spring scales (newton meter) |
Chapter II — Moments
⚖️ Moment of a Force
A force doesn't just push — it can also rotate things. The turning effect of a force is called its moment.
📖 Moment of a Force
The moment of a force = force × perpendicular distance from the force to the fulcrum (pivot point). Unit: N m.
Formula
Moment = F × d
F = force (N) d = perpendicular distance to the fulcrum (m)
🔧 Torque (Moment of a Couple)
📖 A Couple
A couple is two equal but opposite parallel forces acting on an object (e.g. turning a steering wheel).
Torque (Moment of a Couple)
Torque = F × d (between the two forces)
📏 Two Laws of Equilibrium
First Law
The vector sum of all forces acting on a body in equilibrium is zero. (Total forces up = Total forces down)
Second Law
The sum of the clockwise moments about any point equals the sum of the anticlockwise moments.
🧠 How to identify clockwise vs anticlockwise: Imagine the object is pinned at the fulcrum and the only force acting is the one you're looking at. Would it rotate clockwise or anticlockwise? That's your answer!
🎯 Common mistake: The distance in F × d is the distance from the FORCE to the FULCRUM — not from the force to the end of the ruler or to another force. Always measure from force to pivot!
Chapter II — Worked Examples
📐 Calculations: Chapter II
Energy, pressure, gravity and moments — worked step by step.
Example 1 — Kinetic Energy → Height
Q: A 0.005 kg stone is thrown upward at 15 m s⁻¹. Find the maximum height.
Use conservation of energy: KE → PE at the top
½mv² = mgh (mass cancels!)
½(15)² = (9.8)h
½(225) = 9.8h
h = 112.5/9.8 = 11.48 m
½(15)² = (9.8)h
½(225) = 9.8h
h = 112.5/9.8 = 11.48 m
✅ Maximum height = 11.48 m
Example 2 — Power
Q: A crane lifts a 7000 N load to a height of 4 m in 20 s. Find the power output.
Work done = Force × displacement
W = 7000 × 4 = 28,000 J
Power = Work / Time
P = 28,000 / 20 = 1,400 W
✅ Power = 1,400 W
Example 3 — Boyle's Law
Q: Balloon volume = 2 m³ at P = 1000 hPa. Finds volume at P = 500 hPa (constant temperature).
P₁V₁ = P₂V₂
(1000)(2) = (500)(V₂)
V₂ = 2000/500 = 4 m³
V₂ = 2000/500 = 4 m³
✅ New volume = 4 m³ (halving pressure doubles volume)
Example 4 — Gravity on Moon
Q: Find g on the Moon. Radius = 1.7 × 10⁶ m, Mass = 7 × 10²² kg.
Use g = GM/d²
g = (6.7 × 10⁻¹¹)(7 × 10²²) / (1.7 × 10⁶)²
g = (4.69 × 10¹²) / (2.89 × 10¹²)
g = 1.62 m s⁻²
g = (4.69 × 10¹²) / (2.89 × 10¹²)
g = 1.62 m s⁻²
✅ g on Moon ≈ 1.6 m s⁻²
Example 5 — Moments
Q: Metre stick suspended at 20 cm mark. 2 N weight at 15 cm. Weight W acts at 50 cm. Find W.
Clockwise moments = Anticlockwise moments
The 2N force is at 15cm, fulcrum at 20cm → distance = 5cm = 0.05m (anticlockwise)
W is at 50cm, fulcrum at 20cm → distance = 30cm = 0.30m (clockwise)
W is at 50cm, fulcrum at 20cm → distance = 30cm = 0.30m (clockwise)
Set them equal
W × 0.30 = 2 × 0.05
W = 0.10/0.30 = 0.33 N
W = 0.10/0.30 = 0.33 N
✅ W = 0.33 N
Chapter II — Quiz
✅ Quiz: Chapter II
Test yourself on Energy, Pressure, Gravity and Moments!
Chapter III — Vectors
➡️ Vectors
Scalars just have size. Vectors have size AND direction. This distinction unlocks all of mechanics!
| Scalars (magnitude only) | Vectors (magnitude AND direction) |
|---|---|
| Mass, distance, speed, energy, time, power, temperature | Displacement, velocity, acceleration, force, momentum, weight |
🔀 Resolving a Vector into Components
Any vector at an angle can be split into two perpendicular components (usually horizontal and vertical).
Resolving a Vector
Horizontal: v_H = v cos θ
Vertical: v_V = v sin θ
v = magnitude of vector θ = angle to the horizontal
🧠 Memory trick: cos is for the adjacent (horizontal) side. sin is for the opposite (vertical) side. cos = close to horizontal, sin = shooting up!
🔗 Adding Two Perpendicular Vectors
To find the resultant of two perpendicular vectors: use Pythagoras for the magnitude, and tan for the angle.
Resultant Vector
R = √(a² + b²)
θ = tan⁻¹(opposite/adjacent)
⛷️ Inclined Plane
On an inclined plane at angle θ, the component of gravity pulling the object DOWN the slope is:
Component down the slope
F = mg sin θ
Acceleration down slope: a = g sin θ (independent of mass!)
🏹 Projectile Motion
When an object is launched at an angle, treat horizontal and vertical motion separately. The horizontal velocity doesn't change (no force in that direction). The vertical velocity changes due to gravity.
🧠 Key: Horizontal: s = (v cos θ)t (constant speed, no acceleration). Vertical: use UVAST with u = v sin θ and a = −9.8 m s⁻².
Chapter III — Circular Motion
🔄 Circular Motion
Going in a circle at constant speed? You're still accelerating! Here's why — and how to calculate it.
📖 Angular Velocity (ω)
Angular velocity is the rate of change of angle with respect to time. Symbol: ω (omega). Unit: rad s⁻¹.
Formula — Angular Velocity
ω = θ/t v = rω
ω = angular velocity (rad s⁻¹) v = linear speed (m s⁻¹) r = radius (m)
🔵 Why is Circular Motion Accelerating?
Velocity is a vector — it has DIRECTION as well as speed. Even though the speed stays constant, the DIRECTION keeps changing as the object goes around the circle. Changing velocity = acceleration!
📖 Centripetal Force
The force acting towards the centre required to keep an object moving in a circle.
📖 Centripetal Acceleration
The acceleration of a body moving in a circle, directed towards the centre of the circle.
📐 Centripetal Formulas
⚠️ Centripetal Acceleration & Force
a = v²/r = ω²r
F = mv²/r = mω²r
v = linear speed r = radius ω = angular velocity m = mass
Relationship between T and ω
T = 2π/ω ω = 2π/T
T = periodic time (time for one complete circle) (s)
🧵 What provides centripetal force? It's not a new type of force — it's just a name for whatever force is pointing towards the centre. For a ball on a string = tension. For a planet = gravity. For a car on a bend = friction. There is NO "centrifugal force" outward — that feeling is just inertia!
Chapter III — Satellite Orbits
🌙 Satellite Orbits
Kepler figured out the rules of planetary orbits. Newton proved them using his own gravity equations!
🛰️ Deriving the Satellite Period Formula
For a satellite in orbit, the gravitational force provides the centripetal force. By equating Fg = Fc and using v = 2πR/T, we get:
⚠️ Periodic Time of a Satellite
T² = 4π²R³ / GM
T = orbital period (s) R = orbital radius from CENTRE of planet (m)
G = 6.7 × 10⁻¹¹ N m² kg⁻² M = mass of the planet (kg)
G = 6.7 × 10⁻¹¹ N m² kg⁻² M = mass of the planet (kg)
Orbital Velocity (not in tables — need to know!)
v² = GM/R
📡 Geostationary Satellites
📖 Geostationary Satellite
A geostationary satellite remains over one fixed position on Earth. It orbits at the same rate as Earth rotates → periodic time = 24 hours, height ≈ 36,000 km above equator.
Uses: telecommunications, weather satellites, GPS. They're always in the same position in the sky, so a fixed satellite dish can receive their signal.
🎯 Key exam point: r in the satellite formula is the distance from the CENTRE OF EARTH to the satellite. Always add the Earth's radius (6.37 × 10⁶ m) to the altitude above the surface!
Chapter III — SHM
〰️ Simple Harmonic Motion (SHM)
SHM describes the back-and-forth motion of pendulums, springs, and vibrating strings. It's all about restoring forces!
📖 Definition — SHM
An object moves with simple harmonic motion if its acceleration towards a fixed point is directly proportional to its displacement from that fixed point.
Mathematical Expression for SHM
a = −ω²x
a = acceleration (m s⁻²) ω = angular frequency (rad s⁻¹) x = displacement (m)
The minus sign means acceleration is always towards the centre (opposite to displacement)
The minus sign means acceleration is always towards the centre (opposite to displacement)
📊 SHM at Equilibrium vs Extremes
| Position | Displacement | Velocity | Acceleration | Force |
|---|---|---|---|---|
| Equilibrium (centre) | 0 (zero) | maximum | 0 (zero) | 0 (zero) |
| Extreme position | maximum (amplitude) | 0 (zero) | maximum | maximum |
🧠 Why is velocity zero at the extremes? The object has stopped for an instant before reversing direction — just like a ball thrown upward stops at the top.
🧠 Why is acceleration maximum at the extremes? The restoring force is greatest when the displacement is greatest (F ∝ x). More force = more acceleration!
⏱️ Period and Frequency
Period and Frequency of SHM
T = 2π/ω f = ω/(2π) T = 1/f
T = period (time for one oscillation) (s) f = frequency (Hz)
Chapter III — Hooke's Law
🌀 Hooke's Law & Springs
Stretch a spring and it pulls back. The harder you stretch, the harder it pulls. That's Hooke's Law!
📖 Hooke's Law
When a force is exerted to extend or compress a spring, the restoring force is proportional to the displacement (extension or compression). F = −ks
⚠️ Hooke's Law
F = −ks
F = restoring force (N) k = spring constant (N m⁻¹) s = extension (m)
🧠 Hooke's Law IS SHM! A spring obeys F = −ks. From F=ma: ma = −ks → a = −(k/m)s. Compare to a = −ω²s and we get: ω = √(k/m). So any spring executes SHM!
📐 Key Formula for Spring+Mass SHM
⚠️ Angular Frequency of Spring
ω = √(k/m)
k = spring constant m = mass on spring (kg)
Period of Spring Oscillation
T = 2π/ω = 2π√(m/k)
⚖️ At Equilibrium: mg = k(extension)
When a mass hangs on a spring at rest, the weight pulling down = the spring force pulling up:
Equilibrium Condition
mg = k × extension
Use this to find k (if you know the extension) or the extension (if you know k)
⚠️ Important Distinction
- In F = −k(extension): extension = stretch from original natural length
- In a = −ω²s: s = distance from the EQUILIBRIUM POSITION (not the natural length)
🎯 Maximum acceleration occurs at maximum displacement (amplitude). Maximum velocity occurs at the equilibrium position (where acceleration = 0, velocity = 0 at extremes).
Chapter III — Worked Examples
📐 Calculations: Chapter III
Vectors, circular motion, satellites, SHM and Hooke's Law — all worked step by step.
Example 1 — Circular Motion: Force & Radius
Q: Hammer thrower (7.26 kg) swings in a circle, period T = 0.8 s, radius = 2 m. Find (i) ω, (ii) centripetal acceleration, (iii) kinetic energy.
Part (i): ω = 2π/T
ω = 2π/0.8 = 7.85 rad s⁻¹
Part (ii): a = ω²r
a = (7.85)² × 2 = 61.6 × 2 = 123.4 m s⁻² towards centre
Part (iii): v = rω, then KE = ½mv²
v = (2)(7.85) = 15.7 m s⁻¹
KE = ½(7.26)(15.7)² = 895 J
KE = ½(7.26)(15.7)² = 895 J
✅ ω = 7.85 rad s⁻¹, a = 123.4 m s⁻², KE = 895 J
Example 2 — Satellite: Finding Mass of a Planet
Q: Phobos orbits Mars: radius = 9.4 × 10⁶ m, period = 7.6 hours. Find the mass of Mars.
Convert T to seconds
T = 7.6 × 60 × 60 = 27,360 s
Rearrange T² = 4π²R³/GM for M
M = 4π²R³ / (GT²)
M = 4π²(9.4×10⁶)³ / [(6.7×10⁻¹¹)(27,360)²]
M ≈ 6.5 × 10²³ kg
M = 4π²(9.4×10⁶)³ / [(6.7×10⁻¹¹)(27,360)²]
M ≈ 6.5 × 10²³ kg
✅ Mass of Mars ≈ 6.5 × 10²³ kg
Example 3 — Spring: Finding k and Period
Q: Spring natural length 25 mm. Object mass 20 g attached. (i) Find new length. (ii) Find period of oscillation. (k = 12 N m⁻¹)
Part (i): At equilibrium mg = k × extension
mg = (0.02)(9.8) = 0.196 N
0.196 = 12 × extension
extension = 0.0163 m
New length = 25 + 16.3 = 41.3 mm
0.196 = 12 × extension
extension = 0.0163 m
New length = 25 + 16.3 = 41.3 mm
Part (ii): ω = √(k/m), then T = 2π/ω
ω = √(12/0.02) = √600 = 24.5 rad s⁻¹
T = 2π/24.5 = 0.256 s
T = 2π/24.5 = 0.256 s
✅ New length = 41.3 mm, Period = 0.256 s
Example 4 — SHM: Max Force & Frequency
Q: Body (250 g) vibrates: a = −16s. Amplitude = 5 cm. Find (i) frequency, (ii) max force.
From a = −ω²s, identify ω
ω² = 16 → ω = 4 rad s⁻¹
Find frequency: f = ω/(2π)
f = 4/(2π) = 0.637 Hz
Max force: at max displacement (amplitude = 0.05 m)
a_max = ω²s_max = 16 × 0.05 = 0.8 m s⁻²
F_max = ma = (0.25)(0.8) = 0.2 N
F_max = ma = (0.25)(0.8) = 0.2 N
✅ f = 0.637 Hz, F_max = 0.2 N
Chapter III — Quiz
✅ Quiz: Chapter III
Test yourself on Vectors, Circular Motion, Satellites and SHM!
📋 All Formulas — Quick Reference
Every formula you need for the Mechanics Leaving Cert exam.
Chapter I — Motion & Forces
Equations of Motion
v = u + at s = ut + ½at² v² = u² + 2as s = (u+v)t/2
Newton's Second Law
F = ma Weight: W = mg
Momentum & Impulse
ρ = mv Ft = mv − mu
Chapter II — Energy, Pressure, Gravity
Work & Energy
W = Fs EK = ½mv² EP = mgh
Power & Efficiency
P = W/t = Fv % Efficiency = (P_out/P_in) × 100
Density & Pressure
ρ = m/V P = F/A ΔP = ρgΔh P₁V₁ = P₂V₂
Gravity
F = Gm₁m₂/d² g = GM/d²
Moments
Moment = F × d (perpendicular distance to fulcrum)
Chapter III — Advanced Mechanics
Vectors
Horizontal: v_H = v cosθ Vertical: v_V = v sinθ
Circular Motion
v = rω a = v²/r = ω²r F = mv²/r = mω²r T = 2π/ω
Satellite Orbits
T² = 4π²R³/GM v² = GM/R
SHM
a = −ω²x T = 2π/ω f = ω/2π T = 1/f
Hooke's Law & Springs
F = −ks ω = √(k/m) T = 2π√(m/k) mg = k(extension)
G (Universal)
6.7×10⁻¹¹
N m² kg⁻²
g (Earth)
9.8 m s⁻²
N kg⁻¹ (or m s⁻²)
R (Earth)
6.37×10⁶ m
M (Earth)
6.0×10²⁴ kg
📖 All Definitions — Learn These Off!
These are word-for-word definitions for the exam. Memorise the key words!
Chapter I — Motion & Forces
Displacement
Displacement is the straight-line distance between two points, in a given direction.
Velocity
Velocity is the rate of change of displacement with respect to time.
Acceleration
Acceleration is the rate of change of velocity with respect to time.
Acceleration due to Gravity (g)
The acceleration due to gravity is the acceleration of objects in free fall near the Earth's surface due to the gravitational attraction of the Earth. Value: 9.8 m s⁻².
Force
A force is anything which has the potential to cause an object to accelerate. Unit: newton (N).
The Newton
One newton is the force which causes a body of mass 1 kg to undergo an acceleration of 1 m s⁻².
Newton's First Law
Every object will continue to travel at constant velocity unless an unbalanced external force acts on it.
Newton's Second Law
The rate of change of an object's momentum is directly proportional to the force which caused it, and takes place in the direction of the force.
Newton's Third Law
When objects interact, they exert equal but opposite forces on each other.
Momentum
Momentum = mass × velocity. Symbol: ρ. Unit: kg m s⁻¹.
Principle of Conservation of Momentum
In any interaction, the total momentum before equals the total momentum afterwards, provided no external forces act on the system.
Impulse
Impulse is the change in momentum. Impulse = Ft = mv − mu.
Friction
Friction is a force which opposes the relative motion between two objects.
Chapter II — Energy, Pressure, Gravity
Work
Work is defined as the product of force and displacement. Unit: joule (J).
Energy
Energy is the ability to do work. Unit: joule (J).
Principle of Conservation of Energy
Energy cannot be created or destroyed — it can only be converted from one form to another.
Power
Power is the rate at which work is done. Unit: watt (W). 1 W = 1 J s⁻¹.
Solar Constant
The solar constant is the energy from the Sun falling on each square metre of the Earth's surface per second. Unit: W m⁻².
Density
Density is defined as mass per unit volume. ρ = m/V. Unit: kg m⁻³.
Pressure
Pressure is defined as force per unit area. P = F/A. Unit: pascal (Pa). Pressure is a scalar quantity.
Archimedes' Principle
When an object is immersed in a fluid, the upthrust it experiences is equal to the weight of the displaced fluid.
Law of Flotation
A floating object displaces a weight of fluid equal to its own weight.
Boyle's Law
At constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure.
Newton's Law of Universal Gravitation
Any two point masses attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Moment of a Force
The moment of a force = force × perpendicular distance from the force to the fulcrum. Unit: N m.
Chapter III — Advanced Mechanics
Angular Velocity (ω)
Angular velocity is the rate of change of angle with respect to time. Unit: rad s⁻¹.
Centripetal Force
The force acting towards the centre required to keep an object moving in a circle.
Centripetal Acceleration
The acceleration of a body moving in a circle, directed towards the centre of the circle.
Simple Harmonic Motion (SHM)
An object moves with simple harmonic motion if its acceleration towards a fixed point is directly proportional to its displacement from that fixed point.
Hooke's Law
When a force is exerted to extend or compress a spring, the restoring force is proportional to the displacement. F = −ks.
🎯 Exam tip: For definitions, the key words are everything. Words like "rate of change", "proportional", "provided no external forces", "fixed point", "perpendicular distance" are the ones that earn marks. Missing one key phrase can cost you half the marks!