Station 01 · Forces and further motion in a straight line
Acceleration, forces and vectors
When acceleration depends on position, use v dv/dx. When it depends on velocity, separate the variables. Forces are vectors: add them, resolve them and project them.
- dv/dt, v dv/dx, d/dx(½v²)
- a in terms of x, a in terms of v
- Newton's three laws
- F = ma
- resolving forces, projections in 2D and 3D
Acceleration as a function of x
Acceleration as a function of v
Concurrent forces
Drag the force tips. The particle has mass m; the orange line is a direction to project onto.
Projection in 3D
Newton's laws
Station 02 · Simple harmonic motion
Simple harmonic motion
Acceleration a = −n²(x − c): the acceleration always points back to the centre c and grows with distance from it.
- definition and restoring force
- solutions x = A sin(nt + α) + c
- x, v, a graphs
- v² = n²[A² − (x − c)²]
- initial conditions
- proving motion is SHM
- equations from graphs
Displacement x = A sin(nt + α) + c
Verify the solution and derive v²
From initial conditions
Is it simple harmonic?
Equation from a graph
Station 03 · Modelling motion without resistance
Inclined planes and pulleys
Resolve forces along and perpendicular to the plane. Connected particles share one acceleration; write F = ma for each and solve.
- equations of motion
- smooth inclined plane
- single smooth pulley
- hanging mass with a horizontal or inclined plane
Pulley over an inclined plane
Equations
Equations of motion
A block alone on a smooth slope
Station 04 · Rectilinear resisted motion
Resistance ∝ vᵖ
With no other forces, a = −kvᵖ. The power p decides whether the particle stops in finite time, in a finite distance, or never.
- derive a from ΣF = ma
- v as a function of t
- v as a function of x
- x as a function of t
Derivation
Station 05 · Vertical resisted motion
Up, down and terminal velocity
Gravity and air resistance. Going up, both act downwards; coming down, resistance opposes gravity until terminal velocity is reached.
- R ∝ v or v²
- v as a function of t and x
- x as a function of t
- terminal velocity
- maximum height and time to reach it
- time to return
Results
Derivations
Station 06 · Projectiles and resisted motion
Projectiles with resistance
Resistance proportional to velocity acts against both components. The path is no longer a parabola: it rises less, falls more steeply, and approaches a vertical asymptote.
- shape of the trajectory with and without resistance
- equations for acceleration, velocity and displacement
- solving problems
Results
Deriving the equations
Station 07 · Practice
Practice and HSC-style questions
Take g = 9.8 m/s² unless told otherwise. Give decimal answers to 2 decimal places, or exact values.