Mechanics

Mechanics applies mathematical models to describe and predict the motion of objects under the influence of forces. A-Level Mechanics covers kinematics, Newton’s laws, moments, energy, and momentum — the foundations of classical physics that also underpin engineering and applied mathematics.

Topics Covered

Kinematics

• SUVAT equations — constant acceleration: $v = u + at$, $s = ut + \frac{1}{2}at^2$, $v^2 = u^2 + 2as$, $s = \frac{(u+v)}{2}t$
• Graphs of motion — displacement-time, velocity-time, acceleration-time; gradients and areas
• Calculus-based kinematics — $v = \frac{ds}{dt}$, $a = \frac{dv}{dt}$; integration from acceleration to velocity to displacement
• Vertical motion under gravity — $a = -g \approx -9.8\,\text{m/s}^2$

Forces and Newton’s Laws

• Newton’s three laws — inertia, $F = ma$, action-reaction pairs
• Resolving forces — horizontal and vertical components; $F_x = F\cos\theta$, $F_y = F\sin\theta$
• Equilibrium — resultant force equals zero; triangle and polygon of forces
• Friction — $F \leq \mu R$; limiting friction, static vs. dynamic
• Connected particles — pulleys, tow bars, lifts; treating systems and subsystems

Moments

• Moment of a force — $\text{moment} = F \times d$ (force $\times$ perpendicular distance from pivot)
• Principle of moments — sum of clockwise moments = sum of anticlockwise moments for equilibrium
• Centres of mass — uniform laminas, composite bodies
• Tilting and toppling — determining the critical point where an object begins to topple

Energy and Work

• Work done — $W = F \times d$ (in the direction of force); work done against friction and gravity
• Kinetic energy — $KE = \frac{1}{2}mv^2$; gravitational potential energy $PE = mgh$
• Conservation of energy — $KE_1 + PE_1 = KE_2 + PE_2 + W_{\text{friction}}$
• Power — $P = \frac{dW}{dt} = Fv$

Momentum

• Conservation of momentum — $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ (in the absence of external forces)
• Impulse — $I = mv - mu = F\Delta t$; impulse-momentum principle
• Direct collisions — elastic and inelastic impacts

Study Tips

1. Always draw a diagram — label all forces, choose positive directions, and resolve consistently. Marks are lost when directions are ambiguous.
2. Check units — all quantities should be in SI units (metres, kilograms, seconds, newtons) before substituting into equations.
3. Use SUVAT systematically — write down which variables you know and which you need, then select the correct equation.
4. Practise connected particle problems — decide when to treat the system as a whole and when to isolate individual particles.
5. Verify with common sense — if your answer says a car has a negative mass or an object accelerates at $1000\,\text{m/s}^2$, something is wrong.

How to Use These Notes

Follow the sidebar order. Each page provides definitions, derivations, worked examples with full force diagrams, and exam-style problems. Start with kinematics, then forces, then moments and energy.