Vu's Maths Hub: HSC Mechanics: SHM, Resisted Motion, and Calculus-Based Dynamics

Intro

The HSC Mechanics booklet is a free Mathematics Extension 2 resource with 85 worked problems covering simple harmonic motion, variable forces, resisted motion, terminal velocity, projectile motion with resistance, and force analysis. It is written for Extension 2 students mastering calculus-based mechanics; teachers and tutors and designed for structured HSC revision on Vu's Maths Hub.

This deep-dive introduces HSC Mechanics — browser-readable, aligned with the NESA syllabus.

Summary

The HSC Mechanics booklet offers a fundamentals review, 85 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Mechanics — no account required. Use this post to plan how to work through a 85-page booklet efficiently.

What is this booklet?

This booklet is written for Extension 2 students mastering calculus-based mechanics; teachers and tutors.

Focus: simple harmonic motion, variable forces, resisted motion, terminal velocity, projectile motion with resistance, and force analysis.

Topics covered:

Simple harmonic motion
Variable forces
Resisted motion
Terminal velocity
Projectile motion with resistance
Force analysis
Golden-rule acceleration forms

How to use it:

Read Mechanics Primer first
Part 1 before solutions; Part 2 hints after attempt
Revisit hard problems; watch sign conventions
Tutors: Part 1 as worked examples, Part 2 as homework

Approximately 85 pages, CC BY 4.0, readable at HSC Mechanics.

Key fundamental reviews

Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:

Mechanics primer: displacement, velocity, acceleration
Golden-rule acceleration forms ( $\ddot{x}$ , $v \frac{d v}{d x}$ )
Force diagrams and Newton's second law
Simple harmonic motion equations
Resisted motion: linear and quadratic drag models
Terminal velocity and limiting behaviour

Why fundamentals matter

Mechanics primer — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Golden-rule acceleration forms — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Force diagrams and Newton's second law — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Simple harmonic motion equations — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Resisted motion — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Mechanics fundamentals as a closed-book quiz first.

Problems and how to use them

The HSC Mechanics booklet packs 85 practice problems into roughly 85 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.

Overall structure

Part 1 — detailed solutions: basic, medium, and advanced mechanics with detailed solutions.

Part 2 — hint-based fluency: matching tiers with hints after attempt.

Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.

Part 1 (15 problems)

Advanced (5 problems)

This tier contains 5 problems aimed at extension and synthesis. Representative work includes "A bar magnet is held vertically. An object that is repelled by the magnet is …" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

A bar magnet is held vertically. An object that is repelled by the magnet is to be dropped from dire…
An object of mass 1 kg is projected vertically upwards with an initial velocity of u m/s . It experi…
A particle is undergoing simple harmonic motion with period pi 3 . The central point of motion of th…
The point P is 4 metres to the right of the origin O on a straight line. A particle is released from…
A particle of mass m is attracted towards the origin by a force of magnitude mu m x^2 for x neq 0 , …

Basic (5 problems)

This tier contains 5 problems aimed at foundational fluency. Representative work includes "A particle is moving along a straight line. Initially its displacement is at …" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

A particle is moving along a straight line. Initially its displacement is at x = 1 , its velocity is…
The acceleration of a particle is given by x = 32x(x^2 + 3) , where x is the displacement of the par…
A particle is projected from the origin with initial velocity u to pass through a point (a,b) . Prov…
Two model airplanes race around a circular course, with the second airplane taking off T seconds aft…
A particle is moving vertically in a resistive medium under the influence of gravity. The resistive …

Medium (5 problems)

This tier contains 5 problems aimed at exam-standard reasoning. Representative work includes "A particle of mass 1 kg is projected from the origin with speed 40 m s^ -1 at…" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

A particle of mass 1 kg is projected from the origin with speed 40 m s^ -1 at an angle 30^ circ to t…
A particle of unit mass moves horizontally in a straight line. It experiences a resistive force prop…
A particle of mass 1 kg is projected from the origin with a speed of 50 m s^ -1 , at an angle of the…
Two particles, A and B , each have mass 1 kg and are in a medium that exerts a resistance to motion …
A particle of unit mass moves in a straight line against a resistance numerically equal to v+v^3 , w…

Part 2 (70 problems)

Advanced (26 problems)

This tier contains 26 problems aimed at extension and synthesis. Representative work includes "A bungee jumper falls from rest with the cord becoming taut at x = a below th…" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

A bungee jumper falls from rest with the cord becoming taut at x = a below the starting point. For x…
The acceleration of a particle is a=k(1-v^2) with k>0 and initial conditions x=0 , v=0 . Show:
A particle moves with acceleration x = k(1 - v^2) where k is a positive constant. Initially, x = 0 a…
A particle moves with resisted motion governed by x = - lambda(c + v) where lambda, c > 0 . Initiall…
A particle is projected vertically upward with speed u under gravity with air resistance gv^2 k^2 .
… and 21 more at this tier in the booklet

Basic (12 problems)

This tier contains 12 problems aimed at foundational fluency. Representative work includes "A particle moves in a straight line with velocity v = 2t - 5 m/s, where t is …" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

A particle moves in a straight line with velocity v = 2t - 5 m/s, where t is in seconds. The particl…
A particle moving in a straight line experiences an acceleration a = -v^2 10 (m/s ^2 ). Initially ( …
A particle moves in simple harmonic motion about the point x = 3 m with amplitude 2 m. When t = 0 , …
A ball is thrown vertically upwards with initial velocity u m/s from ground level. Find expressions …
A particle moves with acceleration x = 6t - 4 m/s². Initially, the particle is at the origin with ve…
… and 7 more at this tier in the booklet

Medium (32 problems)

This tier contains 32 problems aimed at exam-standard reasoning. Representative work includes "A projectile is fired at speed 102 m/s at angle 45° to the horizontal from gr…" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.

A projectile is fired at speed 102 m/s at angle 45° to the horizontal from ground level. Find:
A particle of mass m is projected vertically upwards from the ground with initial velocity u . Air r…
A particle of unit mass moves against resistance R=v+v^3 . Initially x=0 , v=Q>0 . Show:
A parachutist falls under dv dt =g-kv . For 0 le t le10 , k=0.1 ; for t>10 , k=2.0 (parachute open).…
A mass is dropped from rest at height h and experiences resistance Res = mkv^2 (downwards positive).…
… and 27 more at this tier in the booklet

Common patterns across the booklet

Complex numbers: 1 problem — e.g. "A particle is projected from ground level to hit a target at horizontal distance d and height h . If…"
Vectors & geometry: 5 problems — e.g. "Two model airplanes race around a circular course, with the second airplane taking off T seconds aft…"
Mechanics: 67 problems — e.g. "A particle is undergoing simple harmonic motion with period pi 3 . The central point of motion of th…"
Inequalities: 1 problem — e.g. "A ball is dropped from rest at height h above the ground and rebounds to height 3h 4 after bouncing."
Trigonometry: 1 problem — e.g. "A stone is thrown from the top of a building 45 m high with velocity 15 m/s at angle 30° above horiz…"
Other synthesis: 10 problems — e.g. "A bar magnet is held vertically. An object that is repelled by the magnet is to be dropped from dire…"

Standout and less-seen problem types

These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:

A bar magnet is held vertically. An object that is repelled by the magnet is to be dropped from dire…: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
An object of mass 1 kg is projected vertically upwards with an initial velocity of u m/s . It experi…: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
A particle is undergoing simple harmonic motion with period pi 3 . The central point of motion of th…: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
The point P is 4 metres to the right of the origin O on a straight line. A particle is released from…: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
A particle of mass m is attracted towards the origin by a force of magnitude mu m x^2 for x neq 0 , …: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
A bungee jumper falls from rest with the cord becoming taut at x = a below the starting point. For x…: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.

Working through a large booklet

Read Mechanics Primer first
Part 1 before solutions; Part 2 hints after attempt
Revisit hard problems; watch sign conventions
Tutors: Part 1 as worked examples, Part 2 as homework

Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.

Open HSC Mechanics and work steadily — 85 problems is a marathon, not a sprint.

Key appendices

This booklet has no separate appendix files — formulas and takeaways are embedded in solutions and the conclusion.

Use the conclusion section as a pre-exam checklist and keep your class formula sheet nearby while practising.

Key conclusion

Choose the correct acceleration form; draw force diagrams; apply $F = m a$ systematically; master separation of variables and partial fractions; check limiting behaviour; practise regularly.

The booklet's closing section reinforces these habits:

Read Mechanics Primer first
Part 1 before solutions; Part 2 hints after attempt
Revisit hard problems; watch sign conventions
Tutors: Part 1 as worked examples, Part 2 as homework

Revisit HSC Mechanics in the fortnight before trials and redo problems you missed on first pass.

How to study with this booklet

Study Differential Equations first; two weeks SHM and forces; two weeks resisted motion

General principles:

Closed-book first: attempt without notes, then check fundamentals.
Error log: record concept vs algebra vs reading errors.
Spaced repetition: redo missed questions after 3 and 7 days.
Past papers last: fix weak topics here, then sit full papers timed.

Mini-FAQ

Who is the HSC Mechanics booklet for?

Extension 2 students mastering calculus-based mechanics; teachers and tutors studying Mathematics Extension 2 under the NSW HSC.

Should I read solutions before attempting problems?

Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.

Where can I read the booklet online?

Open HSC Mechanics on Vu's Maths Hub — free, no account required.

How many problems are in the booklet?

Roughly 85 practice problems across 85 pages, each with worked solutions.

Is this aligned with NESA?

Topics match Mathematics Extension 2 outcomes for simple harmonic motion, variable forces, resisted motion, terminal velocity, projectile motion with resistance, and force analysis. Confirm scope with your teacher and current NESA documentation.

Common mistakes to avoid

Using $\ddot{x}$ when the chain rule $v \frac{d v}{d x}$ is required
Sign errors on drag force opposing motion
Forgetting initial conditions when integrating acceleration
Confusing SHM parameters ( $ω$ , amplitude, period)
Rushing to advanced tiers before basic fluency — build foundations first.

Practice on Vu's Maths Hub

Open the free HSC Mechanics on Vu's Maths Hub — 85 problems with full worked solutions.

Related resources:

How to use Vu's Maths Hub — Study DE before Mechanics
HSC Differential Equations — Modelling basics for motion

Simple harmonic motion — exam context

In NSW Mathematics Extension 2 examinations, simple harmonic motion routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Mechanics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Mechanics and locate items that stress simple harmonic motion; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Variable forces — exam context

In NSW Mathematics Extension 2 examinations, variable forces routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Mechanics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Mechanics and locate items that stress variable forces; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Resisted motion — exam context

In NSW Mathematics Extension 2 examinations, resisted motion routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Mechanics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Mechanics and locate items that stress resisted motion; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Terminal velocity — exam context

In NSW Mathematics Extension 2 examinations, terminal velocity routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Mechanics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Mechanics and locate items that stress terminal velocity; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Projectile motion with resistance — exam context

In NSW Mathematics Extension 2 examinations, projectile motion with resistance routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Mechanics booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Mechanics and locate items that stress projectile motion with resistance; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Wednesday, 1 July 2026

HSC Mechanics: SHM, Resisted Motion, and Calculus-Based Dynamics