Further Mechanics

Further Mechanics extends the classical mechanics from A-Level Mathematics into more complex and realistic scenarios: projectile motion in two dimensions, circular motion with varying forces, centres of mass for composite bodies, and elastic collisions where energy conservation interacts with momentum conservation.

Topics Covered

Projectile Motion

• 2D kinematics — resolving velocity into horizontal and vertical components
• Trajectory equation — deriving $y = x\tan\theta - \frac{gx^2}{2v^2\cos^2\theta}$
• Range and maximum height — proofs using SUVAT equations
• Motion on inclined planes — resolving along and perpendicular to the slope
• Vectors and parametric approaches — using $\mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2}\mathbf{g}t^2$

Circular Motion

• Angular velocity and angular acceleration — $\omega = \dot{\theta}$, $\alpha = \ddot{\theta}$
• Centripetal acceleration — $a = \frac{v^2}{r} = r\omega^2$; derivation from first principles
• Horizontal and vertical circles — analysing forces at different positions
• Banked tracks and conical pendulums — resolving forces in rotated frames
• Energy methods — combining conservation of energy with circular motion constraints

Centres of Mass and Elastic Collisions

• Centres of mass — laminae, solid bodies, composite shapes; integration methods for continuous distributions
• Toppling vs. sliding — determining the critical angle for stability
• Momentum and impulse — $\mathbf{I} = m\Delta\mathbf{v}$; vector and scalar forms
• Coefficient of restitution — $e = \frac{v_B - v_A}{u_A - u_B}$; perfectly elastic ($e=1$) and perfectly inelastic ($e=0$) collisions
• Oblique collisions — resolving perpendicular and parallel to the line of centres
• Energy in collisions — kinetic energy lost; when and why conservation fails

Study Tips

1. Draw clear force diagrams — label every force, resolve into components, and choose coordinate axes wisely (often along and perpendicular to the surface or motion direction).
2. Derive the trajectory equation from scratch — it is a common exam request and tests whether you understand the physics, not just the formula.
3. Practise circular motion problems with both horizontal and vertical circles — the force analysis differs significantly between the two.
4. Use conservation laws systematically — for collisions, always check both conservation of momentum and the coefficient of restitution equation. Two equations, two unknowns.
5. Check energy balance — after solving a collision problem, verify that kinetic energy is conserved (if elastic) or correctly reduced (if inelastic).

How to Use These Notes

Follow the sidebar order. Each page contains derivations from first principles, worked examples with full force diagrams, and exam-style problems. Start with projectile motion, then circular motion, then collisions.