Applications of Calculus to Mechanics

mechanics year12 ext2

Applications of calculus to mechanics | NSW Curriculum Website

• ME2-12-05 uses mechanics to model and solve practical problems

📖 Prior Knowledge

Content	Prerequisite relationships
Introduction to Differentiation	- Apply the chain rule → Express acceleration three ways
Further Graph Transformations and Modelling	- Transform trigonometric graphs → Graph SHM
Differential Calculus	- Differentiate composite trig functions → Verify SHM solutions - Differentiate composite trig functions → Prove motion is SHM
Applications of Calculus	- Find velocity and acceleration → Express acceleration three ways - Connect displacement, velocity and acceleration → Newton’s laws of motion
Introduction to Vectors	- Project one vector onto another → Resolve forces into components - Derive projectile equations of motion → Trajectory shapes with resistance
Further Applications of Calculus	- Solve $\frac{dy}{dx}=g(y)$ → Resisted velocity as a function of time - Solve by separation of variables → Motion with velocity-dependent acceleration

Forces and further motion in a straight line

• Derive expressions for acceleration: $\frac{dv}{dt}$, 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 $x=A\sin (nt+\alpha )+c$, $x=A\cos (nt+\alpha )+c$, or $x=A\cos (nt+\alpha )+B\sin (nt+\beta )$ 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 $x=-n^{2}x-c$, 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 $x=A\cos (nt+\alpha )+c$ or $x=A\sin (nt+\alpha )+c$
• 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, $\sum F=mx$, 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, $\sum F=mx$, 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