Applications of Calculus to Mechanics (MEX-M1)

5 dot points across 5 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How does a resistance proportional to speed change rectilinear motion, and what is the terminal velocity of a body falling under gravity with resistance?

Model rectilinear motion under gravity with a resistive force proportional to velocity or to the square of velocity, and determine terminal velocity

Resisted motion. Newton's second law with resistance proportional to speed, separating variables for velocity, the concept of terminal velocity, and motion of a rising and falling body, with verified worked examples.

15 min read →

How do we use Newton's laws to turn the forces acting on a body, including weight, tension, normal and resistive forces, into its equation of motion?

Use Newton's laws to resolve forces and form the equation of motion: weight, tension, normal and resistive forces, equilibrium of concurrent forces, and the conical pendulum

Forces and Newton's laws. Force as a vector, F = ma, weight, tension, normal and resistive forces, resolving into components, the resultant of concurrent forces, equilibrium, building the equation of motion from a free-body diagram, and the conical pendulum, with every number verified in code.

16 min read →

How does the defining equation of simple harmonic motion lead to its displacement, velocity and period, and how do we model an oscillating particle?

Derive and apply the equations of simple harmonic motion, relating acceleration, velocity, displacement, amplitude and period for an oscillating particle

Simple harmonic motion. The defining equation, displacement and velocity functions, amplitude and period, and the velocity-displacement relation, with verified worked examples.

15 min read →

How do we analyse rectilinear motion when acceleration is given as a function of displacement or of velocity rather than of time?

Apply calculus to rectilinear motion where acceleration is expressed as a function of displacement or velocity, using the forms

a = v\,dv/dx

and

a = d(v^2/2)/dx

Rectilinear motion. Acceleration as a function of displacement or velocity, the forms a = v dv/dx and a = d(v^2/2)/dx, and integrating to recover velocity and position, with verified worked examples.

14 min read →

How is two-dimensional projectile motion analysed with calculus, and how does air resistance proportional to velocity change the horizontal and vertical motion?

Analyse projectile motion using calculus, resolving into horizontal and vertical components, and extend to projectiles experiencing a resistance proportional to velocity

Projectile motion. Resolving into independent horizontal and vertical equations, projectile motion without resistance, and resistance proportional to velocity in two dimensions, with verified worked examples.

15 min read →