Module 01 — Mechanics & Motion

Duration: 3–4 weeks | Prereq: Module 00

Mechanics is the foundation of classical physics — the study of how things move, what makes them move, and what happens when they collide. It's the oldest part of physics (Newton published his laws in 1687) and the part most tied to everyday experience. Almost everything in later modules builds on it.

1. Kinematics — Describing Motion

Kinematics is the description of motion without asking why it happens. Save the "why" for Newton's laws in section 2.

Key quantities

Quantity	Symbol	Unit	Scalar/Vector
Displacement	s or x	m	Vector
Distance	d	m	Scalar
Speed	—	m/s	Scalar
Velocity	v	m/s	Vector
Acceleration	a	m/s²	Vector
Time	t	s	Scalar

Displacement vs Distance: Walk 10 m east, then 10 m west. Distance = 20 m. Displacement = 0 m (you're back where you started).

The SUVAT equations

These five equations describe uniform acceleration (acceleration that doesn't change). They're derived from the definitions of velocity and acceleration, but you need to be able to use them fluently.

The five variables: s (displacement), u (initial velocity), v (final velocity), a (acceleration), t (time)

Each equation omits one variable — choose the one that matches what you know and what you want:

Equation	Missing variable
v = u + at	s
s = ut + ½at²	v
v² = u² + 2as	t
s = ½(u + v)t	a
s = vt − ½at²	u

Strategy: List what you know and what you want. Pick the equation that contains those variables. Rearrange if needed.

Worked Example: A car accelerates from rest at 3 m/s² for 8 seconds. How far does it travel?

Known: u = 0, a = 3 m/s², t = 8 s. Want: s.

Use s = ut + ½at²: s = (0)(8) + ½(3)(8²) = 0 + ½(3)(64) = 96 m

Motion graphs

Three graph types, each with specific meanings:

Displacement-time (s-t) graph:

Gradient = velocity
Straight line = constant velocity
Curved line = changing velocity (acceleration)
Horizontal line = stationary

Velocity-time (v-t) graph:

Gradient = acceleration
Area under graph = displacement
Straight line = constant acceleration
Horizontal line = constant velocity

Acceleration-time (a-t) graph:

Area under graph = change in velocity
Horizontal line = constant (uniform) acceleration

Projectile motion

A ball thrown horizontally from a cliff is the classic example. The key insight:

Horizontal and vertical motion are independent.

Horizontally: no force acts (ignoring air resistance), so horizontal velocity is constant. Vertically: gravity acts, so the ball accelerates downward at g = 9.81 m/s².

Worked Example: A ball rolls off a table 1.2 m high at 3 m/s horizontally. Where does it land?

Vertical: s = ½gt² → 1.2 = ½(9.81)t² → t² = 0.2446 → t = 0.495 s

Horizontal: x = 3 × 0.495 = 1.48 m from the base of the table

2. Newton's Laws of Motion

Three laws. All of classical mechanics follows from them.

First Law (Inertia)

An object remains at rest, or moves at constant velocity, unless acted upon by a net external force.

This is profound and counterintuitive. The natural state of things is not to be stationary — it's to keep doing what they're already doing. Friction is the reason things seem to slow down; without it, they wouldn't.

Second Law (F = ma)

The net force on an object equals its mass times its acceleration: F = ma

Units: force in newtons (N) = kg × m/s²

Force is a vector — both magnitude and direction matter
"Net force" means the resultant of all forces acting
If net force = 0, acceleration = 0 (First Law is a special case of this)

Worked Example: A 1200 kg car experiences a driving force of 4000 N and a friction force of 1600 N. What is its acceleration?

Net force = 4000 − 1600 = 2400 N

a = F/m = 2400 / 1200 = 2 m/s²

Third Law (Action-Reaction)

For every force, there is an equal and opposite reaction force acting on a different object.

Critical point that trips people up: action-reaction pairs act on different objects. They don't cancel each other out (cancellation only happens when two forces act on the same object).

Example: You push a wall with force F. The wall pushes you back with force F in the opposite direction. These forces act on different objects (you and the wall). If you're standing on a frictionless surface and push the wall, the wall's reaction force on you is what accelerates you backwards — the two forces don't cancel because they act on different objects. (In practice, a fixed wall transfers the force into the Earth, so the Earth accelerates imperceptibly in the opposite direction.)

3. Forces in Practice

Weight and normal force

Weight is the gravitational force on an object: W = mg (where g = 9.81 m/s² on Earth's surface)

Weight is a vector pointing toward the centre of the Earth.

Normal force (sometimes called normal reaction) is the contact force a surface exerts on an object, perpendicular to that surface. It exists because of Newton's Third Law — the object pushes on the surface, the surface pushes back.

On a flat horizontal surface with no other vertical forces: N = W = mg

Friction

Static friction: prevents an object from starting to move. Has a maximum value. Kinetic (sliding) friction: acts on an object already moving. Always less than maximum static friction.

For kinetic friction: f = μₖN (an equality — friction equals this value while sliding)

For static friction: f ≤ μₛN (an inequality — friction can be anywhere from zero up to this maximum, depending on what's needed to prevent motion)

Where μ is the coefficient of friction (dimensionless, depends on the surfaces) and N is the normal force.

Forces on a slope (inclined plane)

Resolving weight along and perpendicular to the slope is a classic vector problem.

For a slope at angle θ:

Component of weight along the slope (pulling down): mg sin θ
Component of weight perpendicular to slope: mg cos θ
Normal force: N = mg cos θ

[Maths Reminder] This is the component method from Module 00. The weight vector is being resolved into two perpendicular components using sin and cos.

Terminal velocity

When an object falls through air, drag increases with speed. At some point, drag force = weight, net force = 0, acceleration = 0, and the object falls at constant speed — terminal velocity.

The v-t graph for a falling object shows:

Steepening curve as it accelerates under gravity
Levelling off as drag increases
Horizontal line at terminal velocity

4. Work, Energy, and Power

Work

Work is done when a force moves an object in the direction of the force.

W = Fs cos θ

Where F is the force, s is the displacement, and θ is the angle between them.

If force and displacement are in the same direction (θ = 0), then cos 0 = 1 and W = Fs.

Units: Joules (J) = N × m

Key point: If a force acts perpendicular to motion (like the normal force on a ball rolling on a flat surface), it does no work. This is why the normal force doesn't speed things up.

Kinetic and Potential Energy

Kinetic energy (energy of motion): Eₖ = ½mv²

Gravitational potential energy (energy stored by height): Eₚ = mgh

Where h is the height above a reference point.

Conservation of Energy

Energy cannot be created or destroyed, only transformed between forms.

This is one of the most powerful principles in physics. In a closed system (no energy losses):

Eₖ + Eₚ = constant

Worked Example: A ball of mass 0.5 kg is dropped from 10 m. What is its speed just before hitting the ground?

At top: Eₚ = mgh = 0.5 × 9.81 × 10 = 49.05 J, Eₖ = 0

At bottom: Eₖ = 49.05 J (all Eₚ has converted to Eₖ), Eₚ = 0

½mv² = 49.05 → v² = 2 × 49.05 / 0.5 = 196.2 → v = 14 m/s

You get the same answer using v² = u² + 2as. Conservation of energy and kinematics are consistent — they're just different routes to the same physics.

Efficiency

No real system converts energy perfectly. Efficiency tells you how much useful energy you get out:

Efficiency = useful output energy / total input energy (multiply by 100 for %)

Power

Rate of doing work (or transferring energy):

P = W / t (power = work done / time)

Also: P = Fv (useful when something moves at constant speed)

Units: Watts (W) = J/s

5. Momentum

Linear momentum

p = mv (momentum = mass × velocity)

Momentum is a vector. Units: kg m/s

Conservation of momentum

In any collision or explosion, the total momentum of a closed system is conserved (remains constant).

This is Newton's Third Law at work over time. It applies to all collisions — elastic and inelastic.

Total momentum before = Total momentum after

Worked Example (collision): Car A (1000 kg, 20 m/s east) hits stationary car B (1500 kg) and they stick together. What is their velocity after?

Before: p = 1000 × 20 + 1500 × 0 = 20,000 kg m/s

After: p = (1000 + 1500) × v = 2500v

2500v = 20,000 → v = 8 m/s east

Impulse

Impulse = change in momentum = FΔt

This is why airbags work — by increasing the time over which the force acts, they reduce the peak force (same change in momentum, longer time = smaller force).

Elastic vs inelastic collisions

Type	Momentum	Kinetic energy
Elastic	Conserved	Conserved
Inelastic	Conserved	Not conserved (some lost to heat/sound)
Perfectly inelastic	Conserved	Maximum KE lost; objects stick together

Common misconception: Momentum is always conserved in collisions. Kinetic energy is only conserved in perfectly elastic collisions (rare in practice; approximate for gas molecules and subatomic particles).

6. Circular Motion

When something moves in a circle at constant speed, its direction is constantly changing — so its velocity is changing — so it's accelerating. That acceleration points toward the centre of the circle.

Centripetal acceleration: a = v²/r = ω²r

Centripetal force: F = mv²/r = mω²r

Where ω is the angular velocity (rad/s), related to speed by v = ωr. Both forms appear in exam questions so both are worth knowing.

Centripetal force is not a separate force — it's whatever force happens to be pointing toward the centre in a given situation (gravity for satellites, tension for a ball on a string, normal force for a car going over a hill).

Key point: If the centripetal force disappears, the object moves off in a straight line tangent to the circle (First Law). Not outward — tangentially.

7. Simple Harmonic Motion (SHM)

SHM is the motion of anything that oscillates back and forth with a restoring force proportional to displacement. Classic examples: a mass on a spring, a pendulum (approximately).

Key terms:

Amplitude (A): maximum displacement from equilibrium
Period (T): time for one complete oscillation (seconds)
Frequency (f): oscillations per second = 1/T (Hertz, Hz)
Angular frequency (ω): ω = 2πf (radians per second)

Defining condition: acceleration ∝ −displacement a = −ω²x (negative because it always opposes displacement)

Displacement equations (choose based on starting position):

If starting at maximum displacement: x = A cos(ωt)
If starting at equilibrium: x = A sin(ωt)

Velocity at displacement x: v = ±ω√(A² − x²)

At x = 0 (equilibrium): v = ±ωA (maximum speed) At x = ±A (amplitude): v = 0 (momentarily at rest)

Energy in SHM: total energy = ½mω²A² (constant); it swaps between kinetic and potential throughout the oscillation.

Period of a mass-spring system: T = 2π√(m/k) Where k is the spring constant (N/m).

Period of a simple pendulum: T = 2π√(L/g) Where L is the pendulum length. This approximation holds for small angles (roughly θ < 15°), where sin θ ≈ θ, giving a restoring force proportional to displacement. For large swings, the period increases and the simple formula no longer applies.

Note that the pendulum period doesn't depend on the mass or the amplitude (for small angles). This is why pendulums made good clocks.

8. Self-Check Questions

A ball is thrown vertically upward at 15 m/s. How high does it go? (g = 9.81 m/s²)
A 2 kg object is on a slope of 30°. What is the component of its weight along the slope?
A 70 kg runner sprints at 8 m/s. What is their kinetic energy?
In a collision, car A (800 kg, 15 m/s) hits car B (1200 kg, 5 m/s) both going in the same direction. They stick together. What is their final velocity?
A satellite orbits at radius r. If it moves to radius 2r, what happens to its orbital speed? (Use F = mv²/r and gravitational force)

Answers:

v² = u² + 2as → 0 = 225 − 2(9.81)s → s = 225/19.62 = 11.5 m
mg sin 30° = 2 × 9.81 × 0.5 = 9.81 N
Eₖ = ½ × 70 × 64 = 2240 J
p before = 800×15 + 1200×5 = 12000 + 6000 = 18000. v = 18000/2000 = 9 m/s
For circular orbit: mv²/r = GMm/r² → v² = GM/r → v ∝ 1/√r. Double r → v decreases by factor √2, so v is divided by √2 ≈ 1.41

Go Deeper

Mechanics connects directly to everything else. Here's where it leads:

Orbital mechanics and gravity → covered in Module 05 (Cosmology). Kepler's laws, escape velocity, and how Newton's laws predict planetary motion.
Special Relativity → Newton's laws break down at speeds approaching light. The corrections are dramatic and beautiful. Module 05 picks this up.
Fluid mechanics — not covered in this guide but fascinating. How do aeroplanes fly? Why does a ball curve? Bernoulli's equation and viscosity.
Lagrangian mechanics — a more powerful reformulation of Newtonian mechanics using energy rather than forces. University-level but conceptually elegant.

Books if you want to go further:

The Feynman Lectures on Physics, Vol. 1 — the single best introduction to mechanics by the greatest physics teacher of the 20th century. Free online at feynmanlectures.caltech.edu
University Physics by Young & Freedman — comprehensive, used at most UK universities, available in libraries

Next: Module 02 — Waves