NSW Syllabuses

# Mathematics Extension 2 Stage 6 - Year 12 Extension 2 - Mechanics MEX-M1 Applications of Calculus to Mechanics

## Outcomes

#### A student:

• MEX12-6

uses mechanics to model and solve practical problems

• MEX12-7

applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems

• MEX12-8

communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument

## Subtopic Focus

The principal focus of this subtopic is to model the mechanics of objects in a variety of situations, with and without resistance. A calculus-based approach is used to develop models of systems and predict the behaviour of objects.

Students develop an appreciation of mathematical representation in the explanation of mechanics, construct models in situations involving resisted forces and develop high-level problem-solving skills.

## Content

• M1.1: Projectiles
• Students:
• solve problems involving projectiles in a variety of contexts (ACMSM111)
• derive and use the Cartesian equation of the trajectory of a projectile, including problems in which the initial speed and/or angle of projection may be unknown
• use parametric equations of a projectile to determine a corresponding Cartesian equation for the projectile (ACMSM104)
• M1.2: Simple harmonic motion
• Students:
• define simple harmonic motion
• define and use dot notation for derivatives with respect to time
• determine if motion is simple harmonic when given equations of motion for either acceleration, velocity or displacement, and describe the motion in terms of amplitude, frequency, period and phase
• given an equation that describes a particle's motion in the form $$x = a \cos(nt + \alpha)$$, determine that $$v = \dot{x} = - a n \sin(nt + \alpha)$$, and $$\dot{v} = \ddot{x} = - a n^2 \cos(nt + \alpha) = -n^2 x$$
• sketch the graphs of $$x$$, $$\dot{x}$$ and $$\ddot{x}$$ as functions of $$t$$ and note the relationships between the zeros, the maximums and the minimums of the three quantities
• note the physical significance of $$a$$, $$n$$ and $$\alpha$$ and review the terms of amplitude, frequency, period and phase
• determine equations for displacement, velocity and acceleration in terms of time, given that a motion is simple harmonic
• derive and use the expressions $$\frac {dv}{dt}$$, $$v \frac {dv}{dx}$$ and $$\frac{d}{dx} \left( \frac12 v^2 \right)$$ for acceleration (ACMSM136)
• given $$\ddot x = f\left(x\right)$$ and initial conditions, derive $$v^2 = g\left(x\right)$$ and the equation for displacement in terms of time and, hence, describe the resultant motion
• use relevant formulae and graphs to solve problems involving simple harmonic motion
• M1.3: Modelling motion without resistance
• Students:
• use Newton’s laws to obtain equations of motion in situations involving motion other than projectile motion or simple harmonic motion
• describe mathematically the motion of particles in situations other than projectile motion and simple harmonic motion
• review the use of the expressions $$\frac{dv}{dt}$$, $$v \frac{dv}{dx}$$ and $$\frac{d}{dx} \left( \frac12 v^2 \right)$$ for acceleration (ACMSM136)
• derive and use the equations of motion of a particle travelling in a straight line with both constant and variable acceleration  (ACMSM114)
• determine force, acceleration, action and reaction (ACMSM133, ACMSM134, ACMSM135)
• M1.4: Resisted motion
• Students:
• derive, from Newton’s Laws of Motion, the equation of motion of a particle moving in a single direction under a resistance proportional to a power of the speed
• derive an expression for velocity as a function of time (where possible)
• derive an expression for velocity as a function of displacement (where possible)
• derive an expression for displacement as a function of time (where possible)
• solve problems involving resisted motion along a horizontal line
• work with the motion of a particle moving upwards in a resisting medium and under the influence of gravity
• derive, from Newton’s Laws of Motion, the equation of motion of a particle moving vertically upwards in a medium, with a resistance $$R$$ proportional to the first or second power of its speed
• derive an expression for velocity as a function of time and for velocity as a function of displacement (or vice versa)
• derive an expression for displacement as a function of time
• solve problems by using the expressions derived for acceleration, velocity and displacement
• work with the motion of a particle falling downwards in a resisting medium and under the influence of gravity
• derive, from Newton’s Laws of Motion, the equation of motion of a particle falling in a medium, with a resistance $$R$$ proportional to the first or second power of its speed
• determine the terminal velocity of a falling particle from its equation of motion
• derive expressions for velocity as a function of time and for velocity as a function of displacement
• derive an expression for displacement as a function of time
• solve problems by using the expressions derived for acceleration, velocity and displacement