11–12Mathematics Extension 2 11–12 Syllabus

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Implementation from 2026

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Year 12

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Applications of calculus to mechanics

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME2-12-05
uses mechanics to model and solve practical problems

Applications of calculus to mechanics

Forces and further motion in a straight line

Derive expressions for acceleration: $\frac{d v}{d t}$ , where $v$ is a function of $t$ , as well as $v \frac{dv}{dx}$ and $\frac{d}{dx} (\frac{1}{2} v^{2})$ , where $v$ is a function of $x$
Solve problems involving velocity and acceleration expressed in terms of displacement, and acceleration expressed in terms of velocity
Examine Newton’s three laws of motion, including force, acceleration, action and reaction under a constant and non-constant force
Find acceleration, $\ddot{x}$ , using the formula $F = m \ddot{x}$ , where $F$ is the force acting on a mass, $m$
Recognise that forces are vector quantities, analyse concurrent forces on a body by resolving them into perpendicular components and use vector projections to determine how much of a given force acts in a given direction in both 2D and 3D contexts

Simple harmonic motion

Define simple harmonic motion as motion in which acceleration is proportional to, and in the opposite direction to, displacement, that is $\ddot{x} = - n^{2} (x - c)$ , where $x$ is the displacement of a particle from the centre of motion at $x = c$ and $\ddot{x}$ is the acceleration
Recognise that the force $F (x)$ acting on a body of mass, $m$ , executing simple harmonic motion according to the equation $\ddot{x} = {- n}^{2} (x - c)$ , is a restoring force which acts towards the centre of motion causing the body to oscillate around the centre of motion
Verify that $x = A \sin (nt + α) + c$ , $x = A \cos (nt + α) + c$ , or $x = A \cos (nt + α) + B \sin (nt + β)$ are solutions to the differential equation defining simple harmonic motion
Describe simple harmonic motion using displacement, velocity, acceleration, force, amplitude and period
Prove that motion is simple harmonic by obtaining an equation of the form $\ddot{x} = - n^{2} (x - c)$ , when given an equation for acceleration, velocity or displacement
Graph $x$ , $\dot{x}$ and $\ddot{x}$ as functions of $t$ with and without graphing applications for a particle moving in simple harmonic motion where $x$ is of the form $x = A \cos (nt + α) + c$ or $x = A \sin (nt + α) + c$
Determine equations for simple harmonic motion when given graphs of acceleration, velocity or displacement in terms of time
Derive $v^{2} = n^{2} [A^{2} - {(x - c)}^{2}]$ for a particle moving in simple harmonic motion according to $\ddot{x} = {- n}^{2} (x - c)$ , where $A$ is the amplitude
Determine equations for the displacement, $x$ , and velocity, $v$ , in terms of time for an object executing simple harmonic motion according to a given equation $\ddot{x} = {- n}^{2} (x - c)$ and satisfying given initial conditions
Model and solve problems involving simple harmonic motion using relevant formulas and graphs

Modelling motion without resistance

Derive the equations of motion for a particle travelling, without resistance, in a straight line with constant and variable acceleration and use the equations of motion to solve problems
Analyse and solve problems relating to motion on a smooth inclined plane by resolving forces into components parallel and perpendicular to the inclined plane
Solve motion problems involving a single smooth pulley and a smooth inclined plane where a body hangs vertically or lies on a smooth horizontal or inclined plane

Rectilinear resisted motion

Derive, from Newton’s laws of motion, $\sum F = m \ddot{x}$ , the equation for acceleration of a particle moving in a straight line and in the absence of external forces, except for a resistance oppositely directed to the motion and with a magnitude proportional to a power of the speed
Derive an expression for velocity as a function of time of a particle moving in a straight line and in the absence of external forces, except for a resistance oppositely directed to the motion and with a magnitude proportional to a power of the speed
Derive an expression for velocity as a function of displacement of a particle moving in a straight line and in the absence of external forces, except for a resistance oppositely directed to the motion and with a magnitude proportional to a power of the speed
Derive an expression for displacement as a function of time of a particle moving in a straight line and in the absence of external forces, except for a resistance oppositely directed to the motion and with a magnitude proportional to a power of the speed
Solve problems, excluding those with pulley systems, involving a particle moving in a straight line subject to a resistance oppositely directed to the motion and with a magnitude proportional to a power of the speed

Vertical resisted motion

Derive, from Newton’s laws of motion, $\sum F = m \ddot{x}$ , the equation for acceleration of a particle moving vertically (upwards or downwards) in a resistive medium and under the influence of constant gravity, where the particle experiences a resistance, $R$ , oppositely directed to the motion, whose magnitude is proportional to the first or second power of the speed
Derive expressions for velocity both as a function of time and of displacement for vertical resisted motion under the influence of constant gravity, with a resistance, $R$ , oppositely directed to the motion, whose magnitude is proportional to the first or second power of the speed
Derive an expression for displacement as a function of time for vertical resisted motion under the influence of constant gravity, with a resistance, $R$ , oppositely directed to the motion, whose magnitude is proportional to the first or second power of the speed
Define the terminal velocity of a particle falling through a medium as the constant velocity the particle reaches when the resistance of the medium prevents further acceleration
Determine the terminal velocity of a falling particle from its equation of motion for vertical resisted motion under the influence of constant gravity, with a resistance, $R$ , oppositely directed to the motion, and whose magnitude is proportional to the first or second power of the speed
Solve vertical resisted motion problems using the expressions derived for acceleration, velocity and displacement, including finding the maximum height reached by a particle projected vertically upwards and the time taken to reach this maximum height, and finding the time taken for a particle to return to the level from which it was projected and its terminal velocity

Projectiles and resisted motion

Distinguish between the shape of the trajectory of a projectile moving under the influence of gravity and with negligible resistance and the shape of the trajectory of a projectile moving under the influence of gravity subject to resistance whose magnitude is proportional to the speed
Establish and use the equations for acceleration for a projectile moving under the influence of gravity, projected at an angle to the horizontal, and subject to a resistance whose magnitude is proportional to the speed, to solve problems

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11–12Mathematics Extension 2 11–12 Syllabus

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