Modern Physics — practice questions

A muon's internal clock measures its lifetime as $t_0 = 2.20\ \mu\text{s}$. A lab observer sees the muon moving at $v = 0.60c$.

Calculate the Lorentz factor $\gamma$ and the lifetime measured in the lab.

A spacecraft clock runs for $t_0 = 5.0$ s as measured on board. An Earth-based observer measures the same interval as $t = 8.33$ s.

Calculate the speed of the spacecraft relative to Earth, as a fraction of $c$.

Two observers disagree about how long an event lasts, yet both agree on the speed of light.

Explain, starting from Einstein's postulates, why this is not a contradiction, and why proper time is always the shortest time measured for a given pair of events.

A metre stick has a proper length of $1.00$ m. It moves past an observer at $v = 0.60c$, so $\gamma = 1.25$.

Calculate the length the observer measures.

A spacecraft's proper length is $80.0$ m. An Earth observer measures its length as $48.0$ m as it flies past.

Calculate the spacecraft's speed as a fraction of $c$.

A cylindrical spacecraft travels horizontally at high speed. Explain what an observer on Earth measures for its length and its diameter, and explain why these two dimensions are affected differently.

A muon has a proper lifetime of $2.2\ \mu\text{s}$ and travels at $v = 0.98c$, giving $\gamma = 5.03$.

Calculate the muon's lifetime as measured by an observer on Earth.

A muon travels at $v = 0.98c$ ($\gamma = 5.03$) toward the ground, $15\,000$ m away as measured from Earth.

Calculate the distance to the ground as measured in the muon's own frame, and hence show it is plausible for the muon to survive the trip given its proper lifetime of $2.2\ \mu\text{s}$.

Explain, using both the Earth frame and the muon's frame, why more muons reach the ground than classical physics predicts, and explain why the two explanations must give the same result.

State the equivalence principle, and state one observable consequence of it.

Explain why the equivalence principle implies that light must be bent by gravity.

Compare the time dilation predicted by special relativity with that predicted by general relativity, and explain how the GPS system demonstrates the connection between them.

Light of frequency $6.00\times10^{14}$ Hz falls on a metal with work function $W_0 = 1.90$ eV.

Calculate the photon energy in eV, and state whether photoelectrons are emitted.

A metal's threshold frequency is $f_0 = 5.50\times10^{14}$ Hz. Light of frequency $7.20\times10^{14}$ Hz is incident on it.

Calculate the work function of the metal in eV, and the stopping voltage for the emitted electrons.

The intensity of light incident on a metal surface is doubled, while its frequency (above threshold) is kept constant. Explain, using the photon model, what happens to the photoelectric current and to the maximum kinetic energy of the emitted electrons — and explain why a wave model of light could not predict this result correctly.

An electron has momentum $p = 5.00\times10^{-24}$ kg m s⁻¹.

Calculate its de Broglie wavelength.

A proton ($m = 1.67\times10^{-27}$ kg) is accelerated from rest through a potential difference of $500$ V, gaining kinetic energy $E_k = eV = 500$ eV $= 8.01\times10^{-17}$ J.

Calculate the proton's speed and its de Broglie wavelength.

Explain how electron diffraction provides evidence for wave–particle duality, and explain why this effect is never observed for everyday macroscopic objects such as a thrown ball.

An electron drops from an energy level of $-1.5$ eV to $-3.4$ eV, emitting a photon.

Calculate the energy of the emitted photon in eV.

A hydrogen atom has energy levels including $E_2 = -3.40$ eV and $E_4 = -0.85$ eV. An electron absorbs a photon and moves from $E_2$ to $E_4$.

Calculate the frequency of the absorbed photon.

Explain why a hot gas produces bright emission lines at exactly the same frequencies as the dark absorption lines produced when the same (cooler) gas is placed in front of a continuous light source, and explain why only these specific frequencies appear rather than a continuous spectrum.

A nucleus has a mass defect of $\Delta m = 0.0250$ u.

Calculate the binding energy in MeV, using $1$ u $= 931.5$ MeV/c².

A lithium-7 nucleus (mass $7.0154$ u) contains 3 protons ($1.0073$ u each) and 4 neutrons ($1.0087$ u each).

Calculate the binding energy per nucleon, in MeV.

Explain, in terms of the strong nuclear force and mass–energy equivalence, why the mass of a stable nucleus is always less than the sum of the masses of its separate protons and neutrons, and explain what the binding energy physically represents.

In a nuclear reaction, the total mass before is $240.1173$ u and the total mass after is $240.0968$ u.

Calculate the energy released, in MeV.

A uranium-235 nucleus (mass $235.0439$ u) absorbs a neutron ($1.0087$ u) and fissions into barium-141 ($140.9144$ u), krypton-92 ($91.9262$ u), and 3 free neutrons ($1.0087$ u each).

Calculate the Q-value of this reaction in MeV, and state what happens to this energy.

Explain why fission of a heavy nucleus such as uranium-235 releases energy, and explain how the free neutrons produced allow the process to sustain itself as a chain reaction.

In a fusion reaction, the total mass before is $4.0282$ u and the total mass after is $4.0248$ u.

Calculate the energy released, in MeV.

Using the binding energy per nucleon curve, explain why fusing two light nuclei into a heavier one releases energy, referring to how binding energy per nucleon changes.

Using the binding energy per nucleon curve, explain why BOTH fission of a heavy nucleus and fusion of light nuclei release energy, and explain why fusion or fission of nuclei near iron (A ≈ 56) would not release energy.

State one similarity and two differences between the strong interaction and the Coulombic repulsion inside a nucleus.

Explain why stable heavy nuclei contain considerably more neutrons than protons, while stable light nuclei contain roughly equal numbers.

Explain how the competition between the strong interaction and Coulombic repulsion accounts for the shape of the binding energy per nucleon curve, and hence for the fact that both fission and fusion can release energy.

State the quark composition of a proton and of a neutron, and show that each gives the correct total charge.

Explain what happens to the quarks inside a neutron during beta-minus decay, and explain how electric charge is conserved.

Explain how the quark model connects to the strong interaction and to nuclear stability, and discuss why quarks are never observed in isolation.