engineering INTERACTIVE LEARNING PLATFORM

MECHANICS

MILES

Interactive A-Level Mechanics 1 — Master classical mechanics through calculators, simulators, and quizzes.

01 model_training

Mathematical Models

Modelling assumptions, SI units, and the language of mechanics.

02 speed

Constant Acceleration

SUVAT equations, kinematics graphs, vertical motion.

03 north_east

Vectors

Vector operations, magnitude, direction, position vectors.

04 rocket_launch

Dynamics

Newton's laws, F=ma, connected particles, pulleys.

05 swap_horiz

Forces & Friction

Resolving forces, inclined planes, friction.

06 compare_arrows

Momentum & Impulse

Conservation of momentum, collisions, impulse.

07 balance

Statics

Equilibrium, force triangles, Lami's theorem.

08 rotate_right

Moments

Turning effects, beam balance, tilting problems.

model_training CHAPTER 01

THE ART OF MATHEMATICAL

MODELLING IN MECHANICS

Translating real-world problems into mathematical form.

account_tree The Modelling Engine

Real-world Problem

Set Up Mathematical Model

Solve the Mathematical Problem

Reasonable?

YES → Report Results

NO → Reconsider Model

straighten SI Units

BASE UNITS

Masskg

Lengthm

Times

DERIVED UNITS

Speed

Acceleration

Force (Newton)

monitor_weight Mass vs Weight

MASS

check Intrinsic property
check Scalar quantity
check Measured in kg
check Constant everywhere

WEIGHT

check Gravitational force
check Vector (downward)
check Measured in Newtons
check
check Changes with g

warning Kilograms do NOT measure weight. Weight is a force, measured in Newtons.

calculate Weight Calculator

Mass (kg)

Planet

checklist Modelling Assumptions

Objects

Term	Meaning
Particle	Dimensions negligible; mass concentrated at a single point
Rod	Rigid, 1D; does not bend or flex
Lamina	Flat 2D surface; thickness negligible
Uniform body	Mass distributed evenly; centre of mass at geometric centre
Light object	Mass is zero (or negligible)

Environments

Term	Meaning
Smooth surface	No friction
Rough surface	Friction acts on objects
Air resistance	Usually neglected unless stated
Gravity	unless stated otherwise

Connections

Term	Meaning
Inextensible string	Does not stretch; acceleration is the same throughout
Smooth pulley	No friction; tension is the same on both sides
Bead	Particle threaded on a wire; can slide freely
Peg	Fixed support point; string passes over it freely

category Force Taxonomy

FORCES

Field Forces

Weight

Contact Forces

Normal Reaction Friction Tension Thrust Buoyancy Air Resistance

quiz Chapter 1 Quiz

Q1: A ball is modelled as a particle. What does this mean?

a) It's very small

b) Its dimensions are negligible - mass at a point

c) It has no mass

d) It's spherical

Q2: "Smooth surface" means:

a) The surface is flat

b) Friction is zero

c) The surface is horizontal

d) No air resistance

Q3: Weight is measured in:

a) kg

b) m/s²

c) Newtons

d) kg·m

speed CHAPTER 02

CONSTANT

ACCELERATION

The SUVAT equations: your toolkit for constant acceleration problems.

calculate SUVAT Calculator

Check exactly 3 variables as "Known", enter their values, then click CALCULATE.

s (m)

u (m/s)

v (m/s)

a (m/s²)

t (s)

table_chart Equation Decision Matrix

Equation	Missing Variable
	s
	v
	t
	a
	u

show_chart Kinematics Graphs

arrow_upward Vertical Motion

south

Always downward

vertical_align_top

At highest point

north

Up = positive

Convention

quiz Chapter 2 Quiz

Q1: A car accelerates from 5 m/s to 25 m/s in 4s. Find displacement.

a) 60 m

b) 100 m

c) 80 m

d) 50 m

Q2: Which SUVAT equation has no 's'?

a) v = u + at

b) s = ut + ½at²

c) v² = u² + 2as

d) s = ½(u+v)t

Q3: A ball thrown upward at 20 m/s. Time to reach highest point? (g=9.8)

a) 1.0 s

b) 2.04 s

c) 4.08 s

d) 0.5 s

north_east CHAPTER 03

THE GEOMETRY

OF MOTION

compare Scalars vs Vectors

SCALARS

Magnitude only

Distance Speed Mass Time Energy

VECTORS

Magnitude + Direction

Displacement Velocity Acceleration Force Momentum

calculate Vector Calculator

Vector A

i

j

Vector B

i

j

explore Magnitude & Direction Calculator

i component

j component

timeline Position Vector Tracker

r0 (i)

r0 (j)

v (i)

v (j)

Time: 0 s

crisis_alert Collision Detector

Particle A

Pos i

Pos j

Vel i

Vel j

Particle B

Pos i

Pos j

Vel i

Vel j

quiz Chapter 3 Quiz

Q1: The magnitude of the vector 5i + 12j is:

a) 17

b) 13

c) 7

d) 169

Q2: If r = (2i + 3j) + (4i - j)t, the velocity vector is:

a) 2i + 3j

b) 6i + 2j

c) 4i - j

d) 2i - j

Q3: Two particles collide if they are at the same position at the same:

a) Time

b) Speed

c) Place (regardless of time)

d) Acceleration

rocket_launch CHAPTER 04

DYNAMICS OF

A PARTICLE

Newton's laws in action. Force, mass, and acceleration.

inventory_2 Forces Inventory

Weight (W)

W = mg, always downward

Normal Reaction (R)

Perpendicular to surface

Friction (F)

Opposes motion, F ≤ μR

Tension (T)

Along string, away from object

Thrust

Along rod, pushing

Driving Force (D)

Applied force, in direction of motion

calculate F = ma Calculator

Force F (N)

Mass m (kg)

Acceleration a (m/s²)

Enter any 2 values, leave 1 blank to calculate.

Real-time Explorer

Force: 10 N

Mass: 5 kg

Acceleration: 2.00 m/s²

gavel Newton's Laws

1st Law

An object stays at rest or in uniform motion unless acted upon by a resultant force.

Equilibrium: Resultant = 0

2nd Law

The resultant force is proportional to acceleration.

3rd Law

Every action has an equal and opposite reaction.

Equal magnitude, opposite direction

link Connected Particles Solver

Driving Force (N)

Mass 1 (kg)

Mass 2 (kg)

Friction 1 (N)

Friction 2 (N)

settings Pulley Calculator

Mass A (kg) - heavier

Mass B (kg) - lighter

Pulley

A

B

T

account_tree Master Algorithm

Draw Force Diagram

Choose Positive Direction

Apply F = ma

Solve for Unknown

Use SUVAT if needed

quiz Chapter 4 Quiz

Q1: A 5 kg box has a resultant force of 20 N. Its acceleration is:

a) 100 m/s²

b) 4 m/s²

c) 25 m/s²

d) 15 m/s²

Q2: In a pulley system with masses 6 kg and 4 kg, the acceleration is:

a) 9.8 m/s²

b) 4.9 m/s²

c) 1.96 m/s²

d) 2.45 m/s²

Q3: Newton's Third Law states forces are:

a) Equal and opposite

b) Equal and same direction

c) Unequal and opposite

d) Zero

swap_horiz CHAPTER 05

FORCES &

FRICTION

call_split Vector Splitter (Resolving Forces)

Force F (N)

Angle θ (°)

landscape Inclined Plane Simulator

Mass (kg)

Angle α: 30°

texture Friction Calculator

Mass (kg)

μ (coefficient)

Applied Force (N)

Surface angle (°)

Friction Used 0 / 0 N

0 N F_MAX

account_tree Master Protocol

1. Draw force diagram

2. Resolve perpendicular to motion → find R

3. Use F = μR to find friction

4. Resolve along motion → F = ma

quiz Chapter 5 Quiz

Q1: On an inclined plane at angle α, the component of weight down the slope is:

a) mg cos α

b) mg sin α

c) mg tan α

d) mg

Q2: Maximum friction force is given by:

a) F = mg

b) F = ma

c) F = μR

d) F = μmg sin α

Q3: "Limiting equilibrium" means:

a) On the point of moving, friction at maximum

b) Moving at constant speed

c) Decelerating

d) No friction present

compare_arrows CHAPTER 06

MOMENTUM

& IMPULSE

speed Momentum Calculator

Mass m (kg)

Velocity v (m/s)

bolt Impulse Calculator

I = mv - mu (change in momentum)

Mass m (kg)

Initial velocity u (m/s)

Final velocity v (m/s)

explosion Collision Simulator

Particle 1

Mass (kg)

Velocity (m/s)

Particle 2

Mass (kg)

Velocity (m/s)

Collision Type

flip Wall Rebound Calculator

Mass (kg)

Velocity before (m/s)

Velocity after (m/s)

quiz Chapter 6 Quiz

Q1: Momentum is measured in:

a) N

b) kg m/s (or N s)

c) kg m/s²

d) J

Q2: In a collision where particles coalesce, total momentum is:

a) Conserved

b) Doubled

c) Zero

d) Halved

Q3: A 2kg ball hits a wall at 6 m/s and rebounds at 4 m/s. The impulse is:

a) 4 Ns

b) 16 Ns

c) 20 Ns

d) 12 Ns

balance CHAPTER 07

STATICS OF

A PARTICLE

THE PRIME DIRECTIVE

The sum of all forces equals zero for a particle in equilibrium.

compare Two Methods

Component Resolution

Resolve all forces into horizontal and vertical components.

Force Triangles / Lami's Theorem

For 3 forces in equilibrium: each force divided by the sine of the opposite angle is constant.

calculate Equilibrium Solver

Add forces (magnitude and angle from positive x-axis). Up to 5 forces.

F1: at °

F2: at °

F3: at °

F4: at °

F5: at °

grid_view Surface Matrix

Horizontal + Smooth

Forces: W down, R up

R = mg

Horizontal + Rough

Forces: W down, R up, F opposing motion

R = mg, F ≤ μR

Inclined + Smooth

Forces: W down, R perpendicular to slope

R = mg cos α

Inclined + Rough

Forces: W down, R perp. to slope, F along slope

R = mg cos α, F ≤ μR

quiz Chapter 7 Quiz

Q1: For a particle in equilibrium, the resultant force is:

a) Maximum

b) Zero

c) Equal to weight

d) Constant

Q2: Lami's theorem applies to:

a) 2 forces in equilibrium

b) Any number of forces

c) Exactly 3 coplanar concurrent forces in equilibrium

d) Forces on an inclined plane only

Q3: When resolving forces, we decompose into:

a) Perpendicular components

b) Parallel components

c) Equal parts

d) Random directions

rotate_right CHAPTER 08

MOMENTS: THE TURNING

EFFECT OF FORCE

CORE CONCEPT

Moment = Force times perpendicular distance from pivot

calculate Moment Calculator

Force F (N)

Distance d (m)

Angle θ (°) (90=perp)

straighten Beam Balance Simulator

Click on the beam to add weights. Specify pivot position.

Beam length (m)

Pivot position (m from left)

Weight to add (N)

swap_vert Tilting Calculator

Beam length (m)

Beam mass (kg)

Support A position (m)

Support B position (m)

Extra weight position (m)

Extra weight (N)

verified Equilibrium Conditions

No net translational force

No net turning moment

quiz Chapter 8 Quiz

Q1: The moment of a 10 N force acting at a perpendicular distance of 3 m from a pivot is:

a) 3.33 Nm

b) 30 Nm

c) 13 Nm

d) 7 Nm

Q2: For a beam to be in equilibrium:

a) Sum of CW moments = Sum of ACW moments

b) All forces must be equal

c) No forces can act on it

d) The beam must be horizontal

Q3: When a beam is about to tilt about a support, the reaction at the other support is: