A spaceship moves past Earth at \(v=0.60c\). (a) Compute \(\gamma\). (b) An observer on the ship measures their journey to take \(8.0\,\text{years}\) (proper time). How long does an Earth observer measure?
2. [2]
A metre stick on a rocket flies past at \(v=0.80c\) along its length. What is its length as measured by a stationary observer?
3. [3]
Light of frequency \(8.0\times10^{14}\,\text{Hz}\) strikes a metal with work function \(W=2.4\,\text{eV}\). (a) Calculate the photon energy in eV. (b) Determine the maximum kinetic energy of emitted electrons. (c) Will photoemission occur if the frequency is \(5.0\times10^{14}\,\text{Hz}\)? Justify.
4. [3]
Calculate the de Broglie wavelength of (a) an electron with KE \(=100\,\text{eV}\), (b) a \(0.145\,\text{kg}\) baseball at \(40\,\text{m/s}\). Comment on observability.
5. [2]
In the Bohr hydrogen atom, an electron transitions from \(n=4\) to \(n=2\). (a) Find the energy of the emitted photon (eV). (b) Find its wavelength (nm) and identify the visible-spectrum colour.
6. [3]
A radioactive sample of iodine-131 (\(T_{1/2}=8.0\,\text{days}\)) initially has \(8.0\times10^{16}\) atoms. How many remain after \(24\,\text{days}\)? What fraction has decayed?
Part B: Thinking & Investigation [15 marks]
7. [5]
Muons (\(T_{1/2}=1.56\,\mu\text{s}\) at rest) are produced \(10\,\text{km}\) above Earth and travel at \(v=0.995c\). (a) Compute \(\gamma\). (b) Compare the time-of-flight in the Earth frame with the muon's half-life from Earth's perspective and explain why so many reach the surface. (c) Explain the same observation in the muon's rest frame using length contraction.
8. [5]
In a photoelectric experiment, the stopping potential \(V_s\) varies with frequency \(f\) as \(V_s = (h/e)f - W/e\). Data: at \(f=7.0\times10^{14}\,\text{Hz}\), \(V_s=0.80\,\text{V}\); at \(f=10.0\times10^{14}\,\text{Hz}\), \(V_s=2.05\,\text{V}\). (a) Determine Planck's constant from the data. (b) Determine the work function in eV. (c) State the threshold frequency.
9. [5]
A nuclear power reactor uses U-235 fission, releasing about \(200\,\text{MeV}\) per fission. (a) How many fissions per second produce \(1.0\,\text{GW}\) of thermal power? (b) Calculate the mass of U-235 consumed per day in kg (atomic mass \(\approx 235\,\text{u}\), \(1\,\text{u}=1.66\times10^{-27}\,\text{kg}\)). (c) Compare with the mass of coal needed (chemical energy density \(\approx 30\,\text{MJ/kg}\)) for the same energy.
Part C: Communication [15 marks]
10. [5]
Using a labelled diagram of a photoelectric experiment, explain why three of its features (threshold frequency, instantaneous emission, KE depending on frequency not intensity) cannot be explained by the wave model of light.
A student says: "Special relativity is just an illusion — clocks don't really run slow." Write a corrective response. Reference one experimental verification (e.g., Hafele-Keating, muon decay, GPS) and explain what "real" means in this context.
12. [5]
Compare the Rutherford and Bohr models of the atom. For each: how it was developed, what it explains, and one limitation. End by explaining how quantum mechanics resolved Bohr's main shortcoming.
Part D: Application [15 marks]
13. [5]
GPS satellites carry atomic clocks. Special relativity (orbital speed \(\sim 3.9\,\text{km/s}\)) makes satellite clocks run slow by about \(7\,\mu\text{s/day}\) relative to Earth-bound clocks. If this drift were uncorrected, estimate the position error after one day (light travels \(\sim 300\,\text{m/}\mu\text{s}\)). Discuss the social implications.
14. [5]
Carbon-14 (\(T_{1/2}=5730\,\text{yr}\)) is used to date organic remains. A bone fragment shows a \({}^{14}\text{C}\) activity of \(0.10\) of that of a living organism. (a) Determine the age. (b) Discuss two assumptions that limit the accuracy of carbon dating. (c) State a context in which radiocarbon dating provides important social/historical evidence.
15. [5]
A medical PET scanner relies on positron–electron annihilation, where each annihilation produces two \(0.511\,\text{MeV}\) photons. (a) Use \(E=mc^2\) to verify the photon energy from the rest mass of an electron. (b) State why two oppositely directed photons are produced (conservation principles). (c) Discuss one benefit and one risk of PET imaging in medicine.
📖 Complete Answer Key (Click to reveal)
Q1.
(a) \(\gamma=1/\sqrt{1-0.36}=1/\sqrt{0.64}=1.25\). (b) \(\Delta t = 1.25\cdot 8.0 = 10.0\,\text{years}\).
(a) Energy/fission \(=200\,\text{MeV}=3.20\times10^{-11}\,\text{J}\). \(N = 10^9/3.2e{-}11 = 3.13\times10^{19}\,\text{fissions/s}\). (b) Daily atoms \(=N\cdot 86400 = 2.70\times10^{24}\); mass \(=2.70e24\cdot235\cdot1.66e{-}27 = 1.05\,\text{kg/day}\). (c) Coal: total energy \(=10^9\cdot86400=8.64\times10^{13}\,\text{J}\); coal mass \(=8.64e13/3e7\approx 2.88\times10^{6}\,\text{kg}\) (~2900 t). Roughly 3 million times more mass.
Q10.
Wave model would predict (i) any frequency could eject electrons given enough intensity (no threshold), (ii) energy build-up time would delay emission, (iii) intensity should set KE. Photon model: \(E_{photon} = hf\) — explains all three.
Q11.
Time dilation has been verified by atomic-clock comparisons in flight (Hafele-Keating, 1971), atmospheric muons reaching Earth, and GPS satellite-clock corrections. It is "real" in the operational sense — measurements differ in different frames.
Q12.
Rutherford: scattering experiment showed nucleus is small/dense; explained mass concentration; couldn't explain why orbiting electrons don't radiate and spiral in. Bohr: postulated quantized orbits; explained H spectral lines; ad hoc and failed for multi-electron atoms. Quantum mechanics: replaces orbits with probability distributions (orbitals), naturally explains stability and spectra.
Q13.
Position error \(=300\,\text{m}/\mu\text{s}\cdot7\,\mu\text{s}=2100\,\text{m/day}\) — would render GPS useless without correction. Social implications: navigation, telecommunications, finance time-stamping all rely on relativistic corrections.
Q14.
(a) \(0.10 = (1/2)^{t/5730} \Rightarrow t = 5730\log_2(10) = 19{,}035\,\text{yr}\approx 1.9\times10^4\,\text{yr}\). (b) Assumes constant atmospheric \({}^{14}\text{C}/{}^{12}\text{C}\) and that the sample exchanged carbon with the atmosphere until death; both are imperfect (industrial CO\(_2\) and reservoir effects). (c) Dating archeological remains, validating recorded historical timelines.
Q15.
(a) \(E_0 = m_e c^2 = (9.11e{-}31)(9e16)=8.20e{-}14\,\text{J} = 0.511\,\text{MeV}\). (b) Conservation of momentum (zero before in CM frame) and energy → two photons of equal energy in opposite directions. (c) Benefit: functional imaging of metabolism (cancer, brain). Risk: ionizing-radiation dose; need for short-lived isotopes.