Mechanics

MEX-M1 Applications of Calculus to Mechanics

• 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 AAM

  not written yet · needs a worked example

  • establish that simple harmonic motion is modelled by equations of the form:

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  • establish that when a particle moves in simple harmonic motion about $c$, the central point of motion, then $\ddot{x}=-n^2(x-c)$

    not written yet · needs a worked example

• prove that motion is simple harmonic when given an equation of motion for acceleration, velocity or displacement and describe the resulting motion

  not written yet · needs a worked example

• sketch graphs of $x,\dot{x}$ and $\ddot{x}$ as functions of $t$ and interpret and describe features of the motion

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• 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

  not written yet · needs a worked example

• 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

  not written yet · needs a worked example

• use relevant formulae and graphs to solve problems involving simple harmonic motion AAM

  not written yet · needs a worked example

• examine force, acceleration, action and reaction under constant and non-constant force (ACMSM133, ACMSM134) AAMACMSM133 ACMSM134

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• examine motion of a body under concurrent forces (ACMSM135) AAMACMSM135

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• 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 $\frac{d}{dx}\left(\frac{1}{2}{v}^2\right)$ for acceleration (ACMSM136) AAMACMSM136

  not written yet · needs a worked example

• use Newton’s laws to obtain equations of motion in situations involving motion other than projectile motion or simple harmonic motion AAM

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  • use $F=m\ddot{x}$ where $F$ is the force acting on a mass, $m$, with acceleration $\ddot{x}$

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• describe mathematically the motion of particles in situations other than projectile motion and simple harmonic motion AAM

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  • 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

    not written yet · needs a worked example

• derive and use the equations of motion of a particle travelling in a straight line with both constant and variable acceleration (ACMSM114) AAMACMSM114

  not written yet · needs a worked example

• solve problems involving resisted motion of a particle moving along a horizontal line AAM

  not written yet · needs a worked example

  • 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

    not written yet · needs a worked example

  • derive an expression for velocity as a function of time

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  • derive an expression for velocity as a function of displacement

    not written yet · needs a worked example

  • derive an expression for displacement as a function of time

    not written yet · needs a worked example

  • solve problems involving resisted motion along a horizontal line

    not written yet · needs a worked example

• solve problems involving the motion of a particle moving vertically (upwards or downwards) in a resisting medium and under the influence of gravity AAM

  not written yet · needs a worked example

  • 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

    not written yet · needs a worked example

  • derive an expression for velocity as a function of time and for velocity as a function of displacement (or vice versa)

    not written yet · needs a worked example

  • derive an expression for displacement as a function of time

    not written yet · needs a worked example

  • determine the terminal velocity of a falling particle from its equation of motion

    not written yet · needs a worked example

  • 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

    not written yet · needs a worked example

• solve problems involving projectiles in a variety of contexts AAM

  not written yet · needs a worked example

  • use parametric equations of a projectile to determine a corresponding Cartesian equation for the projectile

    not written yet · needs a worked example

  • 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

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• 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 AAM

  not written yet · needs a worked example