Extension Mechanics – Applications of Calculus

This lesson comprises seven (7) master classes focusing on:

• Simple harmonic motion
• Motion under concurrent forces
• Displacement-time and velocity-time graphs
• Resisted motion
• Terminal velocity

Content:

MEX-M1.1

• Derive equations for displacement, velocity and acceleration in terms of time, given that a motion is simple harmonic and describe the motion modelled by these equations 
  • establish that simple harmonic motion is modelled by equations of the form: \( x=a \cos(nt+ \alpha)+c \) or \( x=a \sin(nt+\alpha)+c \), where \( x \) is displacement from a fixed point, \( a \) is the amplitude, \( \frac{2 \pi}{n} \) is the period, \( frac{\alpha}{n} \) is the phase shift and \( c \) is the central point of motion
  • establish that when a particle moves in simple harmonic motion about \( c \), the central point of motion, then \{ \ddot{x}=−n^2(x−c) \)
• Prove that motion is simple harmonic when given an equation of motion for acceleration, velocity or displacement and describe the resulting motion
• Sketch graphs of \( x \), \( \dot{x} \) and \( \ddot{x} \) as functions of t and interpret and describe features of the motion
• Prove that motion is simple harmonic when given graphs of motion for acceleration, velocity or displacement and determine equations for the motion and describe the resulting motion
• Derive \( v^2=g(x) \) and the equations for velocity and displacement in terms of time when given \( \ddot{x}=f(x) \) and initial conditions, and describe the resulting motion
• Use relevant formulae and graphs to solve problems involving simple harmonic motion

MEX-M1.2

• Examine force, acceleration, action and reaction under constant and non-constant force
• Examine motion of a body under concurrent forces
• Consider and solve problems involving motion in a straight line with both constant and non-constant acceleration and derive and use the expressions \( \frac{dv}{dt} \), \( v \frac{dv}{dx} \) and \( \frac{d}{dx}(\frac{1}{2}v^2) \) for acceleration
• Use Newton’s laws to obtain equations of motion in situations involving motion other than projectile motion or simple harmonic motion
  • use \( F=m \ddot{x} \) where \( F \) is the force acting on a mass, \( m \), with acceleration \( \ddot{x} \)
• Describe mathematically the motion of particles in situations other than projectile motion and simple harmonic motion
  • interpret graphs of displacement-time and velocity-time to describe the motion of a particle, including the possible direction of a force which acts on the particle
• Derive and use the equations of motion of a particle travelling in a straight line with both constant and variable acceleration

MEX-M1.3

• Solve problems involving resisted motion of a particle moving along a horizontal line
  • 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
  • derive an expression for velocity as a function of displacement
  • derive an expression for displacement as a function of time
  • solve problems involving resisted motion along a horizontal line
• Solve problems involving the motion of a particle moving vertically (upwards or 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 moving vertically 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
  • determine the terminal velocity of a falling particle from its equation of motion
  • solve problems by using the expressions derived for acceleration, velocity and displacement including obtaining the maximum height reached by a particle, and the time taken to reach this maximum height and obtaining the time taken for a particle to reach ground level when falling

MEX-M1.4

• Solve problems involving projectiles in a variety of contexts
  • use parametric equations of a projectile to determine a corresponding Cartesian equation for the projectile
  • 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
• Solve problems involving projectile motion in a resisting medium and under the influence of gravity which include consideration of the complete motion of a particle projected vertically upwards or at an angle to the horizontal