Physics Advanced

Kinematics describes motion without asking what causes it. For uniform acceleration a along a straight line, four equations connect displacement s, initial velocity u, final velocity v, and time t:

• v = u + at
• s = ut + ½at²
• v² = u² + 2as
• s = ½(u + v)t

We use g = 9.8 m/s² downward for free-fall problems throughout this course. Displacement and velocity are vector quantities (direction matters); speed and distance are scalar.

A ball is dropped from rest (u = 0) and falls for 3 s.
  v = 0 + 9.8 × 3 = 29.4 m/s
  s = 0 × 3 + ½ × 9.8 × 3² = 44.1 m

Isaac Newton’s three laws are the foundation of classical mechanics:

• First law (Inertia): An object remains at rest or moves at constant velocity unless acted upon by a net external force.
• Second law: F = ma — net force equals mass times acceleration. Units: newtons (N = kg·m/s²).
• Third law (Action–Reaction): For every action force there is an equal and opposite reaction force acting on the other object.

The free-body diagram is an essential tool: draw each force as an arrow on the object to find the net force.

A 5 kg box accelerates at 3 m/s² on a frictionless surface.
  F = ma = 5 × 3 = 15 N

Momentum is defined as p = mv (kg·m/s). It is a vector — direction matches velocity. The law of conservation of momentum states that in a closed system with no external forces, total momentum before a collision equals total momentum after.

Impulse J = FΔt equals the change in momentum: J = Δp = mv − mu. A large force over a short time produces the same impulse as a small force over a longer time — the principle behind airbags and crumple zones.

Two carts collide and stick together (perfectly inelastic):
  Cart A: mass 2 kg, velocity 6 m/s →
  Cart B: mass 4 kg, at rest
  Total momentum before = 2 × 6 + 4 × 0 = 12 kg·m/s
  Combined mass = 6 kg
  Velocity after = 12 ÷ 6 = 2 m/s →

Work done by a constant force: W = Fd cosθ, where θ is the angle between force and displacement. Units: joules (J).

Two forms of mechanical energy:

• Kinetic energy: KE = ½mv²
• Gravitational potential energy: PE = mgh

The work–energy theorem states W_net = ΔKE. In the absence of friction, total mechanical energy is conserved: KE + PE = constant.

Power is the rate of doing work: P = W/t = Fv. Units: watts (W = J/s).

A 2 kg ball dropped from h = 5 m (g = 9.8 m/s²):
  PE at top = 2 × 9.8 × 5 = 98 J
  KE at bottom = 98 J  →  v = √(2 × 98 / 2) = √98 ≈ 9.9 m/s

An object moving at constant speed in a circle has a centripetal acceleration directed toward the centre:

• ac = v² / r
• Period: T = 2πr / v
• Centripetal force: Fc = mv² / r

Centripetal force is not a new type of force — it is whichever real force (tension, gravity, friction, normal force) points inward. The speed is constant but direction changes continuously, so the velocity — and hence acceleration — are not zero.

A 0.5 kg ball on a 1.2 m string moves in a horizontal circle at 4 m/s.
  F_c = mv²/r = 0.5 × 16 / 1.2 ≈ 6.67 N (tension in string)

Temperature measures average kinetic energy of particles. Heat (Q) is energy transferred by thermal contact. The specific heat capacity (c) relates heat to temperature change:

• Q = mcΔT (J), where m is mass in kg and ΔT in °C or K.

The three ideal gas laws (fixed amount of gas):

• Boyle’s law: P₁V₁ = P₂V₂ (constant T)
• Charles’s law: V₁/T₁ = V₂/T₂ (constant P, T in kelvin)
• Combined: P₁V₁/T₁ = P₂V₂/T₂

How much heat to raise 2 kg of water (c = 4 200 J/kg·K) by 10 K?
  Q = 2 × 4 200 × 10 = 84 000 J = 84 kJ

A wave transfers energy without transferring matter. Key quantities:

• Wavelength λ (m): distance between successive crests.
• Frequency f (Hz): oscillations per second.
• Wave equation: v = fλ

Light obeys two fundamental laws of optics:

• Law of reflection: angle of incidence = angle of reflection (both measured from the normal).
• Snell’s law of refraction: n₁ sinθ₁ = n₂ sinθ₂

A converging lens has focal length f; the thin-lens equation is 1/f = 1/do + 1/di.

Sound travels at 340 m/s. Find the wavelength of a 680 Hz note.
  λ = v / f = 340 / 680 = 0.5 m

Ohm’s law: V = IR — voltage (V) equals current (I, in amperes) times resistance (R, in ohms Ω).

Combining resistors:

• Series: Rtotal = R₁ + R₂ + … — same current through each.
• Parallel: 1/Rtotal = 1/R₁ + 1/R₂ + … — same voltage across each.

Electrical power: P = VI = I²R = V²/R (watts).

Energy: E = Pt (joules, or kilowatt-hours for billing).

Two resistors: R₁ = 4 Ω, R₂ = 12 Ω in parallel. Find R_total.
  1/R = 1/4 + 1/12 = 3/12 + 1/12 = 4/12
  R_total = 12/4 = 3 Ω

A magnetic field (B, in teslas T) exerts a force on moving charges: F = qvB sinθ, and on current-carrying wires: F = BIL sinθ.

The direction of the magnetic force on positive charges is given by the right-hand rule: point fingers in the direction of v (or I), curl toward B, and the thumb points in the direction of F.

Faraday’s law of electromagnetic induction: a changing magnetic flux through a loop induces an EMF:

• EMF = −ΔΦ/Δt, where Φ = BA cosθ is magnetic flux (webers, Wb).
• Lenz’s law: the induced current opposes the change in flux.

A 0.5 T field acts on a 2 A wire of length 0.3 m at 90°.
  F = BIL sin90° = 0.5 × 2 × 0.3 × 1 = 0.3 N

The Bohr model pictures the atom as a nucleus (protons + neutrons) surrounded by electrons in discrete energy levels. Electrons emit light of specific wavelengths when they drop to lower energy levels.

Key nuclear quantities:

• Mass number A = protons (Z) + neutrons (N).
• Isotopes: same Z, different N.
• Radioactive decay types: α (helium nucleus), β (electron or positron), γ (high-energy photon).
• Half-life t_½: time for half the nuclei to decay.
• Einstein’s mass–energy equivalence: E = mc² (c ≈ 3 × 10⁸ m/s).

After 3 half-lives, what fraction of a sample remains?
  (½)³ = 1/8 of the original amount