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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=acos(nt+α)+c or x=asin(nt+α)+c, where x is displacement from a fixed point, a is the amplitude, 2πn is the period, fracαn 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, ˙x and ¨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 v2=g(x) and the equations for velocity and displacement in terms of time when given ¨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 dvdt, vdvdx and ddx(12v2) 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¨x where F is the force acting on a mass, m, with acceleration ¨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