Specialist maths mechanics: kinematics in VCE Specialist and NSW Extension 2, with worked questions

The daily Specialist Maths: Mechanics paper has 8 original questions worth 25 to 35 marks. For VCE Specialist, its constant acceleration, velocity-time graph, projectile and variable acceleration questions fit; skip forces, inclined planes, friction, connected particles and moments, which the course doesn't include. Extension 2 students should check each question against the content of the mechanics focus area before sitting it.

Mechanics in VCE Specialist Maths

The 2023 study design puts kinematics in two areas of study. In calculus: rectilinear motion of a single particle, velocity-time graphs, and differentiation, antidifferentiation and differential equations applied to motion, with acceleration written as d²x/dt², dv/dt, v dv/dx or d(½v²)/dx. In space and measurement: vector kinematics in one, two and three dimensions, differentiating and antidifferentiating vector functions of time.

School-assessed Coursework is 20% in Unit 3 and 20% in Unit 4, and both exams come with a formula sheet. In 2026, Exam 1 is on Monday 9 November and Exam 2 on Wednesday 11 November. Specialist Units 3 and 4 assume you are taking, or have taken, Mathematical Methods Units 3 and 4.

Mechanics in NSW Extension 2

The Mathematics Extension 2 Syllabus (2024) has five Year 12 focus areas: the nature of proof, further work with vectors, introduction to complex numbers, further integration, and applications of calculus to mechanics. It is first examined in the 2027 HSC, in a 100-mark paper of 3 hours plus 10 minutes reading time, with 10 marks of multiple choice and a reference sheet.

Specialist maths mechanics questions with answers

We wrote these for this page. Questions 1 to 3 suit a no-technology exam.

1. Rectilinear motion. A particle starts at the origin with velocity v(t) = t² − 6t + 8 m/s. When is it at rest, and how far does it travel in the first 4 seconds? Answer: v = (t − 2)(t − 4), so it is at rest at t = 2 and t = 4. Antidifferentiate: x(t) = t³/3 − 3t² + 8t, so x(2) = 20/3 and x(4) = 16/3. It goes out 20/3 m, then back 4/3 m, so the distance is 8 m, while the displacement is only 16/3 m.

2. Acceleration as a function of position. A particle moves with acceleration a = −4x and passes the origin at 6 m/s. Find v² in terms of x and the furthest it gets from the origin. Answer: write a = d(½v²)/dx = −4x, so ½v² = −2x² + c. At x = 0, v = 6, so c = 18 and v² = 36 − 4x². The particle stops where v = 0, at x = ±3 m: this is simple harmonic motion with amplitude 3.

3. Velocity-time graph. A train speeds up uniformly from rest to 12 m/s in 4 s, holds that speed for 6 s, then slows uniformly to rest in 3 s. How far does it travel? Answer: add the areas of the two triangles and the rectangle under the graph: ½ × 4 × 12 + 6 × 12 + ½ × 3 × 12 = 24 + 72 + 18 = 114 m. Read a velocity-time graph in our Laws of Motion course, then press Mark.

4. A differential equation of motion. A boat cuts its engine at 30 m/s and then slows according to dv/dt = −0.2v. Find v(t) and the time for its speed to halve. Answer: separating variables gives v = 30e^−0.2t. It halves when e^−0.2t = ½, so t = ln 2 ÷ 0.2 ≈ 3.47 s, whatever the starting speed. The Differential Equations Calculator plots the solution.

5. Vector kinematics. A ball's position in metres after t seconds is r(t) = 15t i + (20t − 4.9t²) j, with j vertically up from level ground. Find its greatest height and where it lands. Answer: differentiate: v(t) = 15 i + (20 − 9.8t) j. The vertical component is zero at t = 20 ÷ 9.8 ≈ 2.04 s, where the height is 20t − 4.9t² ≈ 20.4 m. It lands when 20t − 4.9t² = 0, at t ≈ 4.08 s, a horizontal distance of 15 × 4.08 ≈ 61.2 m away. Check it in the Projectile Motion Calculator.

Use the daily paper as a trial

1. Open today's mechanics paper and work to the suggested time, about 40 to 45 minutes.
2. VCE Specialist: do the constant acceleration, velocity-time graph, projectile and variable acceleration questions without technology, as Exam 1 practice; skip the force questions.
3. Extension 2: check each question against your mechanics content points before you start.
4. Mark it, then use past VCAA or NESA papers for full-length practice across every topic.

Common mechanics mistakes

• Giving displacement when the question asks for distance. Find when the particle turns round and add each leg.
• Forgetting the constant when you antidifferentiate velocity or acceleration. Use the initial conditions to find it.
• Using v dv/dx when acceleration depends on time. Pick the form that matches the variable acceleration is given in.
• Mixing up components in vector kinematics: set only the vertical component to zero at the top.
• Dropping the sign in a resistance term. Resistance opposes motion, so it carries a minus sign.

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