Worked examples: HSC Physics Formula Sheet

Formula index with worked examples

1. Motion · constant acceleration

   Formula or reference: s = ut + at²/2. v² = u² + 2as. v = u + at.

   Units and conditions: Use metres, seconds and signed components along the chosen axis.

   Worked example: At u = 2 m/s and a = 1 m/s² for 4 s, v = 6 m/s and s = 16 m. The time-free form gives 6² = 2² + 2 × 1 × 16.

2. Forces · Newton’s second law

   Formula or reference: Fnet = ma, as a vector equation.

   Units and conditions: Force in N, mass in kg and acceleration in m/s².

   Worked example: A net 15 N to the east on a 5 kg mass gives acceleration 3 m/s² east.

3. Energy · work and power

   Formula or reference: W = F∥s = Fs cos θ. P = ΔE/Δt. Instantaneous P = F∥v = Fv cos θ.

   Units and conditions: Energy and work in J, power in W. Use the angle between force and displacement or velocity as appropriate.

   Worked example: A 10 N force along 3 m does 30 J of work. If that takes 2 s, average power is 15 W. At speed 4 m/s along the force, instantaneous power is 40 W.

4. Energy · potential and kinetic

   Formula or reference: ΔU = mgΔh near Earth’s surface. K = mv²/2.

   Units and conditions: Use kg, m and s to obtain J.

   Worked example: A 2 kg object rises 3 m and reaches 4 m/s. Its potential energy rises by 58.9 J at g = 9.81 m/s², and its final kinetic energy is 16 J.

5. Collisions · kinetic energy and momentum

   Formula or reference: Elastic: Σ(mv²/2)before = Σ(mv²/2)after. Isolated system: Σ(mv)before = Σ(mv)after.

   Units and conditions: Momentum uses signed or vector velocities in m/s. Energy is in J and momentum in kg m/s.

   Worked example: In a head-on elastic collision, equal masses can exchange velocities. A 1 kg mass at +3 m/s and a stationary 1 kg mass keep total momentum +3 kg m/s and kinetic energy 4.5 J after swapping velocities.

6. Momentum · impulse

   Formula or reference: Δp = FnetΔt for a constant or average net force.

   Units and conditions: Impulse in N s equals change of momentum in kg m/s.

   Worked example: A net 5 N to the right for 0.4 s adds 2 N s of rightward momentum.

7. Rotation · angular speed and torque

   Formula or reference: ω = Δθ/t. τ = r⊥F = rF sin θ.

   Units and conditions: Use radians and seconds for angular speed. Torque in N m. The torque angle is between the lever arm and force.

   Worked example: A rotation of π radians in 2 s averages π/2 rad/s. A perpendicular 20 N force on a 0.3 m lever produces 6 N m.

8. Motion · circular speed, acceleration and force

   Formula or reference: v = 2πr/T. ac = v²/r. Fc = mv²/r.

   Units and conditions: Use m, s and kg. Centripetal acceleration and net force point inward.

   Worked example: For r = 2 m and T = π s, v = 4 m/s. The inward acceleration is 8 m/s² and the net inward force on 3 kg is 24 N.

9. Gravity · force and potential energy

   Formula or reference: F = GMm/r². U = −GMm/r, taking zero potential energy at infinity.

   Units and conditions: Use centre-to-centre distance in m, mass in kg, force in N and energy in J.

   Worked example: For GMm = 120 J m and r = 4 m, attraction has magnitude 7.5 N and U = −30 J.

10. Orbits · circular period and radius

    Formula or reference: r³/T² = GM/(4π²)

    Units and conditions: Use orbital radius in m and period in s. M is the central mass.

    Worked example: Around the same body, increasing circular orbital radius by a factor of four increases period by 4³⁄² = 8.

11. Waves · speed and period

    Formula or reference: v = fλ. f = 1/T.

    Units and conditions: Use Hz, s, m and m/s.

    Worked example: A 0.25 s period gives f = 4 Hz. With wavelength 1.5 m the wave speed is 6 m/s.

12. Waves · beats

    Formula or reference: fbeat = |f₂ − f₁|

    Units and conditions: Frequency in Hz.

    Worked example: Tones at 440 Hz and 444 Hz produce 4 beats per second.

13. Waves · classical Doppler shift

    Formula or reference: f′ = f(vwave + vobserver)/(vwave − vsource) with positive speeds for approach in the corresponding terms.

    Units and conditions: Speeds relative to the medium in matching units. Frequency in Hz.

    Worked example: A 660 Hz source moving toward a stationary observer at 10 m/s in air with sound speed 340 m/s is heard at 660 × 340/330 = 680 Hz.

14. Waves · diffraction grating

    Formula or reference: d sin θ = mλ

    Units and conditions: d and λ in the same length unit. m is the diffraction order.

    Worked example: For d = 2.0 μm and λ = 500 nm, first-order sin θ = 0.25, so θ ≈ 14.5°.

15. Optics · refractive index, refraction and critical angle

    Formula or reference: n = c/v. n₁ sin θ₁ = n₂ sin θ₂. sin θc = n₂/n₁ when n₁ > n₂.

    Units and conditions: Indices are unitless. Angles are measured from the normal.

    Worked example: If light speed in a medium is 2.0 × 10⁸ m/s, n ≈ 1.50. From that medium to air, 30° incidence gives refraction near 48.6° and the critical angle is about 41.8°.

16. Optics · polarisation and spreading

    Formula or reference: Malus’s law: I = Imax cos² θ. For an isotropic point source: I₁r₁² = I₂r₂².

    Units and conditions: Intensity in W/m², distance in m. θ is the angle between the polarisation direction and analyser axis.

    Worked example: At 60°, a polariser transmits one quarter of the maximum intensity. Doubling distance from an isotropic source also reduces intensity to one quarter, by a different mechanism.

17. Thermal physics · heating and conduction

    Formula or reference: Q = mcΔT. Q/t = kAΔT/d for steady conduction through a slab.

    Units and conditions: Use J, s and K for heat, time and temperature difference, with k in W/(m K).

    Worked example: A 0.2 kg sample with c = 500 J/(kg K) gains 300 J for a 3 K increase. A 0.1 m slab of area 0.5 m², k = 0.2 W/(m K), across 10 K conducts at 10 W.

18. Electric fields · plates, force and potential

    Formula or reference: E = V/d. F = qE as a vector equation. V = ΔU/q. Work magnitude can be qV or qEd for aligned positive magnitudes.

    Units and conditions: Use V, m, N/C, C and J. For signed work use the actual direction and potential change.

    Worked example: Across 0.020 m at 20 V, E = 1000 V/m. A +2.0 μC charge feels 0.0020 N and the field does 40 μJ of work moving it through the gap along the field.

19. Electric forces · Coulomb’s law

    Formula or reference: F = k|q₁q₂|/r², with k = 1/(4πε₀).

    Units and conditions: Use C and m. Determine attraction or repulsion from charge signs.

    Worked example: Two +1.0 μC charges 0.30 m apart repel with magnitude about 0.100 N.

20. Circuits · current, voltage and power

    Formula or reference: I = q/t. V = IR. P = VI.

    Units and conditions: Use C, s, A, V, Ω and W.

    Worked example: If 12 C flows in 3 s, I = 4 A. Through 2 Ω that requires 8 V and dissipates 32 W.

21. Magnetism · straight wire and ideal solenoid

    Formula or reference: Straight wire: B = μ₀I/(2πr). Long solenoid: B = μ₀NI/L.

    Units and conditions: Use A and m to obtain T. N is total turns.

    Worked example: At 0.10 m from a long straight wire carrying 5 A, B = 1.0 × 10⁻⁵ T. With 200 turns over 0.40 m carrying 2 A, an ideal solenoid has B ≈ 1.26 × 10⁻³ T.

22. Magnetism · forces on charges and wires

    Formula or reference: F = |q|vB sin θ for a particle. F = IℓB sin θ for a wire.

    Units and conditions: θ is between velocity or current and B. Use C, m/s, A, m and T to obtain N.

    Worked example: For perpendicular motion at 500 m/s, a 2 μC charge in 0.2 T feels 0.0002 N. A perpendicular 0.5 m wire carrying 2 A in the same field feels 0.20 N.

23. Magnetism · parallel wire force per length

    Formula or reference: F/ℓ = μ₀I₁I₂/(2πr)

    Units and conditions: Use A and separation in m to obtain N/m. Currents in the same direction attract.

    Worked example: Two long parallel wires carrying 2 A and 3 A, 0.10 m apart, attract at 1.2 × 10⁻⁵ N/m.

24. Induction · magnetic flux and EMF

    Formula or reference: Φ = B⊥A = BA cos θ. ε = −NΔΦ/Δt.

    Units and conditions: θ is the angle from the area normal. Flux in Wb and EMF in V.

    Worked example: A field normal to a 0.020 m² loop increases from zero to 0.50 T in 0.10 s. The change is 0.010 Wb per turn, giving |ε| = 1.0 V for 10 turns.

25. Motors · coil torque

    Formula or reference: τ = nIAB sin θ, with θ between the area normal and B.

    Units and conditions: Use A, m² and T to obtain N m.

    Worked example: A 20-turn coil carrying 2 A with area 0.010 m² in a 0.30 T field has maximum torque 0.12 N m.

26. Circuits · ideal transformer

    Formula or reference: Vp/Vs = Np/Ns. VpIp = VsIs.

    Units and conditions: Use consistent AC voltage and current measures.

    Worked example: An ideal 4:1 turns ratio changes 24 V and 0.5 A on the primary to 6 V and 2 A on the secondary.

27. Quantum physics · matter waves and photon energy

    Formula or reference: λ = h/(mv) for nonrelativistic matter. E = hf for a photon.

    Units and conditions: Use h in J s, mass in kg, speed in m/s and frequency in Hz.

    Worked example: A nonrelativistic particle with momentum 3.32 × 10⁻²⁷ kg m/s has wavelength about 200 nm. Light at 5.0 × 10¹⁴ Hz has photon energy about 3.32 × 10⁻¹⁹ J.

28. Quantum physics · photoelectric effect

    Formula or reference: Kmax = hf − φ when hf ≥ φ.

    Units and conditions: Use the same energy unit for photon energy and work function.

    Worked example: An incident photon of 4.0 eV on a 2.5 eV surface can produce an electron with maximum kinetic energy 1.5 eV.

29. Thermal radiation · Wien’s law

    Formula or reference: λmax = b/T

    Units and conditions: Use T in K and b in m K to obtain m.

    Worked example: For T = 5800 K and b = 2.898 × 10⁻³ m K, the peak wavelength is about 500 nm.

30. Atomic spectra · hydrogen wavelengths

    Formula or reference: 1/λ = R(1/nf² − 1/ni²) for emission with ni > nf.

    Units and conditions: R in m⁻¹ and λ in m. The quantum numbers are unitless.

    Worked example: For ni = 3 and nf = 2, the factor is 1/4 − 1/9 = 5/36, giving λ ≈ 656 nm with the hydrogen Rydberg constant.

31. Relativity · mass–energy

    Formula or reference: E = mc² for rest energy.

    Units and conditions: Mass in kg and c in m/s give energy in J.

    Worked example: A mass change of 1.0 × 10⁻¹⁰ kg corresponds to 9.0 × 10⁶ J using c ≈ 3.0 × 10⁸ m/s.

32. Relativity · time, length and momentum

    Formula or reference: t = t₀/√(1 − v²/c²). ℓ = ℓ₀√(1 − v²/c²). p = m₀v/√(1 − v²/c²).

    Units and conditions: Use the proper time t₀ and rest length ℓ₀. Momentum uses rest mass m₀.

    Worked example: At 0.60c the square-root factor is 0.8. A proper 4 s interval is observed as 5 s, a rest length of 10 m contracts to 8 m, and momentum is 1.25m₀v.

33. Nuclear physics · radioactive decay and half-life

    Formula or reference: N(t) = N₀e^(−λt). λ = ln 2/t½.

    Units and conditions: Use the same time units in t and half-life. λ is in inverse time.

    Worked example: With a 4 h half-life, λ ≈ 0.173 h⁻¹. After 8 h the expected remaining count is one quarter of the original.

More worked examples

NESA's Physics syllabus page links the data sheet, formula sheet and periodic table. The exam specifications say these are provided. Practise recognising the physical model and converting quantities into compatible units before substitution.

1. Photon energy

   Relationship: E = hf = hc/λ.

   Calculate the energy of a photon of wavelength 600 nm, using h = 6.63 × 10^−34 J s and c = 3.00 × 10^8 m/s.

   Show worked response and error check

   Convert 600 nm to 6.00 × 10^−7 m. E = (6.63 × 10^−34 × 3.00 × 10^8)/(6.00 × 10^−7) = 3.315 × 10^−19 J, or 3.32 × 10^−19 J to three significant figures.

   Check: Substituting 600 as a length in metres gives the wrong scale. Photon energy is not power.

2. Circular motion

   Relationship: The net inward force has magnitude F = mv²/r for uniform circular motion.

   A 2.0 kg object moves at 3.0 m/s in a circle of radius 1.5 m. Find the required net inward force.

   Show worked response and error check

   F = 2.0 × 3.0² / 1.5 = 12 N, directed towards the centre. In a physical setup, identify which actual force or combination supplies this net force.

   Check: Centripetal force describes the net inward requirement; do not add it as an extra force to a free-body diagram.

3. Average induced EMF

   Relationship: Magnitude of average EMF = N|ΔΦ|/Δt, with Φ = BA cos θ and θ measured from the area normal.

   A 20-turn coil has flux per turn decrease from 0.006 Wb to zero in 0.04 s. Find the average EMF magnitude.

   Show worked response and error check

   |ΔΦ| = 0.006 Wb. EMF = 20 × 0.006 / 0.04 = 3 V. Lenz's law determines the direction by opposing the change in flux, not simply the original field.

   Check: Do not multiply by area again when flux is already supplied in webers. The question asks for average EMF, not an instantaneous value.