Intro
Mechanics questions in Extension 2 ask you to translate a physical situation into vectors and differential equations, choose a sign convention, and solve with initial conditions. A clear diagram with forces labelled beats jumping straight to algebra. Keywords: mechanics modelling, HSC physics/maths integration, motion questions. NSW Extension 2.
Summary
Draw a diagram, define positive direction, list forces, write F = ma or a DE for velocity, apply initial conditions, interpret physically (e.g. terminal velocity). Projectile, resisted motion, and connected particles are standard scenarios.
Mechanics is the application capstone for vectors and differential equations in Extension 2. Students who master DE setup first report mechanics word problems become routine rather than intimidating.
Key Points
- Diagram first: weight, normal, tension, friction, resistance — all labelled.
- Sign convention: pick positive direction and stick to it for every force.
- Newton's second law F = ma links force sum to acceleration.
- Resisted motion often yields dv/dt = k(a − v) or similar separable DE.
- Terminal velocity: set dv/dt = 0 and solve.
- Prerequisites: HSC Differential Equations then HSC Mechanics.
Worked example
Question. A particle falls vertically with resistance R = 0.5v (v in m/s). Taking downward positive and g = 10 m/s², write the DE for velocity and find terminal velocity.
Solution.
- Forces downward: mg. Upward: resistance 0.5v.
- F = ma: mg − 0.5v = m dv/dt. With m = 1: 10 − 0.5v = dv/dt.
- Terminal velocity: dv/dt = 0 → 10 − 0.5v = 0 → v = 20 m/s.
Answer. dv/dt = 10 − 0.5v; terminal velocity 20 m/s downward.
Takeaway. At terminal speed acceleration is zero — solve the equilibrium equation before integrating.
Exam Preparation
Mechanics combines vectors, DEs, and interpretation. Study DE setup first, then mechanics word problems weekly. Always state units and whether velocity is positive upward or downward.
Mechanics revision pairs well with differential equations day-by-day: morning DE setup, afternoon mechanics interpretation. This mirrors the Extension 2 syllabus linkage and builds transferable modelling skills for Q16-style combined questions.
- Force diagram drill. Sketch five scenarios labelling all forces.
- DE setup. Translate word problems to dv/dt = ... without solving.
- Interpret solutions. State terminal velocity and direction in words.
Mechanics questions often specify 'neglect air resistance' except when resistance is given — read the stem twice. Units must be consistent (SI). If velocity changes direction, the DE may require piecewise solutions with different sign conventions. Compare terminal velocity predictions with limiting behaviour of your solution as t → ∞. Practise quoting given values before substituting to show examiners you understand the model.
Mini-FAQ
Do I need Physics for Extension 2 Mechanics?
The maths syllabus defines the models — physics intuition helps but exam marking follows the maths setup you write.
What if resistance is proportional to v²?
Still set up F = ma; separability depends on the exact form — check the DE booklet examples.
Can I use energy methods?
Only if the question allows — HSC marking schemes usually expect Newton/DE approach from the syllabus.
Common mistakes to avoid
- Mixing sign conventions mid-solution.
- Forgetting resistance opposes motion.
- No diagram — leads to missing normal or tension.
- Solving for velocity without applying initial conditions when required.
Create a forces checklist taped to your desk: weight, normal, tension, friction, resistance — tick each force on every diagram before writing equations. Terminal velocity problems often appear as 'show that v approaches …' — verify by setting acceleration to zero before integrating. Pair every mechanics session with one DE setup question from the Differential Equations booklet.
Resisted motion and projectile models dominate Extension 2 mechanics — practise sign conventions until automatic. Displacement from velocity may require integrating the velocity function found from a DE — show each link explicitly.
Quote given values before substituting into DEs — examiners reward explicit modelling steps. If velocity changes sign, reconsider your positive direction choice. Mechanics and DE booklets on vumaths.com are designed to be studied in that order for Extension 2. Terminal velocity is an equilibrium condition: set dv/dt = 0 before integrating.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Mechanics booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
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