Mechanics
What you must understand
Extension 2 mechanics centres on motion with variable acceleration and resisted motion, expressed through differential equations. You must set up equations of motion from Newton's second law, use (a = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d^2x}{dt^2}) appropriately, solve separable differential equations for velocity and displacement, handle resisted motion (proportional to velocity or its square), and analyse projectile motion with or without resistance. Simple harmonic motion - including its differential equation, period, amplitude and phase - is also essential.
Traps
- Choosing the wrong acceleration form: use (v\frac{dv}{dx}) when the force is given as a function of position; use (\frac{dv}{dt}) when it depends on time or velocity directly.
- Sign conventions: pick a positive direction and keep it; gravity and resistance may act oppositely.
- Terminal velocity: arises when acceleration is zero, i.e. driving force balances resistance - solve for it rather than assuming.
- Constants of integration: determine them from initial conditions before substituting further.
- SHM identification: (\ddot{x} = -n^2 x) about the centre; displacement must be measured from the equilibrium point, not an arbitrary origin.
Rule of thumb
Write the equation of motion first, then decide which form of acceleration makes the resulting differential equation separable in the given variables.
Worked example
A particle falls from rest under gravity with resistance proportional to velocity: (\frac{dv}{dt} = g - kv). Separating, [ \int \frac{dv}{g - kv} = \int dt \implies -\frac{1}{k}\ln(g-kv) = t + C. ] With (v=0) at (t=0), (C = -\frac{1}{k}\ln g), giving (v = \frac{g}{k}(1 - e^{-kt})). As (t\to\infty), (v \to \frac{g}{k}), the terminal velocity.