Modern Physics — Free Physics Review Games.

Albert Einstein published the theory of special relativity in 1905.

Einstein's equation E = mc^2 shows that mass and energy are interchangeable, with c being the speed of light.

The photoelectric effect occurs when photons of sufficient energy strike a metal surface and eject electrons.

A photon is a quantum (discrete packet) of electromagnetic energy that exhibits both wave and particle properties.

Radioactive decay is the spontaneous emission of particles or energy from unstable atomic nuclei.

Wave-particle duality is the concept that all matter and energy exhibit both wave and particle characteristics.

Max Planck proposed that energy is quantized, emitted in discrete packets (quanta) with energy E = hf.

Special relativity predicts that a moving clock runs slower compared to a stationary observer, an effect significant near the speed of light.

Nuclear fission splits heavy atoms (like uranium) into lighter fragments, releasing enormous amounts of energy.

The strong nuclear force is the strongest fundamental force, binding protons and neutrons together in the atomic nucleus over very short distances.

According to special relativity, as an object's speed approaches c, the energy required to accelerate it further approaches infinity.

De Broglie proposed that all moving particles have a wavelength given by wavelength = h/(mv), where h is Planck's constant.

Heisenberg's uncertainty principle states that the more precisely you know a particle's position, the less precisely you can know its momentum, and vice versa.

Antimatter consists of antiparticles (like positrons and antiprotons) with opposite charge; when matter meets antimatter, they annihilate and release energy.

E = hf (Planck's equation) shows that photon energy is directly proportional to frequency, with h being Planck's constant.

The Lorentz factor \(\gamma = \frac{1}{\sqrt{1-v^2/c^2}}\) scales how much time dilates, length contracts, and relativistic mass increases for objects moving at high speed relative to an observer, growing without bound as \(v\) approaches \(c\). The choice "The rate at which a radioactive nucleus decays over time" describes the decay constant from nuclear physics, an unrelated concept from a different branch of the unit. Students should remember that \(\gamma\) is the single unifying factor that appears in nearly every relativistic transformation equation.

Length contraction predicts that a stationary observer measures a moving object's length as shorter than its proper length only along the direction of motion, described by \(L = L_0/\gamma\). The option "Shorter, but only perpendicular to the direction of motion" is incorrect because dimensions perpendicular to motion are unaffected by relativistic contraction. The key exam takeaway is that relativistic effects like length contraction and time dilation only apply along the axis of relative motion.

Planck's constant links a photon's energy directly to its frequency through \(E = hf\), showing that light energy comes in discrete quanta rather than a continuous range. The distractor "The mass of a particle to its energy" instead describes Einstein's \(E=mc^2\), a completely separate relation from special relativity. Students should recognize \(h\) as the fundamental proportionality constant connecting quantum energy to oscillation frequency.

The threshold frequency is the minimum frequency of incident light required to give photons enough energy to overcome the metal's work function and eject electrons. The choice "The frequency of the emitted electrons after ejection" confuses light frequency with electron kinetic energy, which is a separate quantity determined by \(KE = hf - \phi\). A key principle is that below threshold frequency, no electrons are emitted regardless of light intensity, showing light's particle nature.

An alpha particle is identical to a helium-4 nucleus, containing two protons and two neutrons bound together and emitted during alpha decay. The distractor "A single high-energy electron" actually describes a beta particle, which is emitted in a different type of nuclear decay. Recognizing the composition of each decay particle helps predict how the parent nucleus's atomic and mass numbers change.

In beta-minus decay, a neutron in the nucleus converts into a proton, emitting an electron and an antineutrino to conserve charge and lepton number. The option "A photon" instead describes gamma decay, where the nucleus releases excess energy without changing its proton or neutron count. Students should link each decay type to the specific particle it emits: alpha releases helium nuclei, beta releases electrons or positrons, and gamma releases photons.

Half-life is defined as the time it takes for exactly half of the radioactive nuclei in a sample to undergo decay, a constant characteristic of each isotope regardless of sample size. The distractor "The total energy released during one decay event" describes the decay energy, an entirely different physical quantity measured in MeV. This concept underlies exponential decay models, where the remaining quantity follows \(N = N_0 (1/2)^{t/T_{1/2}}\).

The rest energy equation shows that mass itself is a form of energy, even when an object is not moving, with \(c^2\) acting as the conversion factor between mass and energy units. The option "The energy released only during nuclear fission" is too narrow because rest energy applies to all matter, not just fissionable nuclei undergoing reactions. This equivalence principle explains why small amounts of mass converted in nuclear reactions release enormous amounts of energy.

Combining \(E = hf\) with the wave relation \(c = f\lambda\) gives \(E = hc/\lambda\), showing that photon energy is inversely proportional to wavelength. The option \(E = h\lambda c\) incorrectly multiplies wavelength instead of dividing, which would predict that longer wavelengths carry more energy, contrary to observation. Students should remember that shorter-wavelength light, like ultraviolet or X-rays, carries more energy per photon than longer-wavelength light like infrared.

Gamma rays are high-energy electromagnetic photons released when a nucleus transitions from an excited state to a lower energy state, carrying no mass or charge. The distractor "A fast-moving helium nucleus" actually describes an alpha particle, a completely different, massive form of radiation. A useful distinction for the exam is that gamma decay changes only the energy state of the nucleus, not its proton or neutron count.

Einstein's second postulate states that the speed of light in a vacuum, \(c\), is measured to be the same constant value by every inertial observer, regardless of their motion or the motion of the light source. The option "Faster for observers moving toward the light source" reflects classical Galilean velocity addition, which special relativity replaced. This postulate is the foundation from which time dilation and length contraction are mathematically derived.

The Bohr model proposed that electrons occupy discrete, quantized energy levels around the nucleus, and they emit or absorb photons only when jumping between these specific levels. The option "Electrons move in continuous, unrestricted orbits of any energy" describes the classical picture Bohr's model replaced, which could not explain the discrete spectral lines observed in hydrogen. This quantization idea was an important early step toward full quantum mechanics.

Isotopes share the same atomic number, meaning the same number of protons, but differ in neutron number, which changes their mass number and often their nuclear stability. The option "They have the same neutrons but different protons" would actually describe atoms of different elements entirely, since the number of protons defines the element. Recognizing isotopes is essential for understanding radioactive decay, since unstable isotopes of an element decay while stable ones do not.

The relativity of simultaneity states that whether two spatially separated events occur at the same time depends on the observer's inertial reference frame, since the finite speed of light means different observers receive signals at different times relative to their own motion. The option "simultaneous for all observers, regardless of motion" reflects the classical Newtonian assumption of absolute time, which special relativity overturned. This concept shows that time itself, not just measurements of it, is frame-dependent in relativity.

Relativistic momentum is given by \(p = \gamma m v\), where the Lorentz factor \(\gamma\) accounts for the increasing resistance to acceleration as velocity approaches the speed of light. The classical formula \(p = mv\) is only an approximation valid at low speeds where \(\gamma \approx 1\), and it fails to predict the divergence of momentum near \(c\). This relativistic correction explains why infinite energy would be required to accelerate a massive object to the speed of light.

Using the relativistic velocity addition formula \(u = \dfrac{v+u'}{1+vu'/c^2}\), combining two \(0.5c\) velocities gives \(u = \dfrac{c}{1.25} = 0.8c\), correctly staying below the speed of light. The option "Exactly \(1.0c\), from simple addition" applies classical Galilean addition, which is invalid at relativistic speeds and would violate the speed-of-light limit. This formula guarantees that combining velocities never produces a result exceeding \(c\), preserving relativity's core postulate.

Because \(KE_{max} = hf - \phi\), increasing the photon frequency directly increases the maximum kinetic energy of ejected electrons while the work function \(\phi\) stays fixed for a given metal. The option "increase the number of electrons ejected per second" is incorrect because electron emission rate depends on light intensity, not frequency, for light above the threshold. This distinction shows that frequency controls electron energy while intensity controls electron quantity in the photoelectric effect.

In Compton scattering, the photon transfers some of its energy and momentum to the electron it strikes, causing the scattered photon to have lower energy and therefore a longer wavelength, consistent with \(E = hc/\lambda\). The option "It becomes zero, since the photon is fully absorbed" describes a different process, photoelectric absorption, not the elastic scattering seen in the Compton effect. This wavelength shift provided strong evidence that photons carry momentum and behave as particles in collisions.

The emitted photon's frequency is set by the energy difference between the two quantized levels through \(f = (E_2 - E_1)/h\), since energy must be conserved during the transition. The option "The atom's temperature at the moment of transition" is irrelevant because individual atomic transitions depend on quantized energy states, not the bulk thermal property of temperature. This relationship explains why atoms produce sharp, characteristic spectral lines rather than a continuous range of emitted frequencies.

The "mass defect" arises because some of the nucleons' mass is converted into binding energy, via \(E = \Delta m c^2\), that holds the nucleus together against electrostatic repulsion between protons. The option "Protons lose their charge when bound in the nucleus" is false, since protons retain their positive charge regardless of nuclear binding. This mass-energy conversion explains why nuclear reactions can release far more energy per reaction than chemical reactions.

Twelve days corresponds to three half-lives (\(12/4 = 3\)), so the remaining fraction is \((1/2)^3 = 1/8\) of the original sample. The option \(\dfrac{1}{16}\) would correspond to four half-lives, or 16 days, which overshoots the given time interval. This exponential decay pattern, where each half-life halves the remaining quantity, is central to solving all radioactive decay problems.

The Sun generates energy through nuclear fusion, in which hydrogen nuclei combine under extreme pressure and temperature to form helium, releasing energy because the helium product has slightly less mass than the combined hydrogen reactants. The option "Fission of heavy uranium nuclei" describes the process used in nuclear power plants on Earth, not the process occurring in stars. Both fusion and fission release energy because they move nuclei toward the more tightly bound configuration near iron on the binding energy curve.

Planck resolved the "ultraviolet catastrophe," where classical physics incorrectly predicted a blackbody would radiate infinite energy at short wavelengths, by proposing that energy is emitted in discrete quanta rather than continuously. The option "The inability to explain why metals conduct electricity" is unrelated, since electrical conduction is explained by band theory in solid-state physics, not blackbody radiation. Planck's quantization hypothesis marked the historical beginning of quantum mechanics as a field.

Because the traveling twin moves at relativistic speed relative to the Earth-bound twin for much of the trip, time dilation causes fewer seconds to pass for the traveler as measured by an Earth clock, resulting in genuinely less biological aging. The option "Time dilation only affects clocks, not biological aging" is incorrect because time dilation affects all physical processes, including biological ones, not just mechanical timekeeping devices. This asymmetry is resolved because the traveling twin, unlike the stationary twin, undergoes acceleration and changes reference frames during the trip.

Critical mass is the minimum quantity of fissile material required so that, on average, at least one neutron from each fission event goes on to trigger another fission, sustaining the chain reaction. The option "It is the mass of neutrons required to start the reaction" incorrectly focuses on neutron mass rather than the total fissile material needed to maintain the reaction. Below critical mass, too many neutrons escape without causing further fissions, and the chain reaction dies out.

The Born interpretation states that \(|\psi|^2\) gives the probability density of locating a particle at a particular position when a measurement is made, reflecting the inherently probabilistic nature of quantum systems. The option "The exact position of the particle at all times" contradicts quantum mechanics, since particles do not have definite positions until measured. This probabilistic interpretation is a foundational departure from classical mechanics' deterministic trajectories.

The relativistic energy-momentum relation \(E^2 = (pc)^2 + (mc^2)^2\) correctly combines a particle's momentum and rest mass into its total energy, reducing to \(E = mc^2\) when \(p=0\) and to \(E \approx pc\) for massless particles like photons. The option \(E = \dfrac{p^2}{2m}\) is the classical nonrelativistic kinetic energy formula, which fails at speeds approaching \(c\). This equation is essential for analyzing high-energy particles and massless photons within a single consistent framework.

Radiocarbon dating measures the remaining fraction of carbon-14 in a once-living sample and uses its known half-life of about 5,730 years to calculate how long ago the organism died and stopped absorbing new carbon-14. The option "The rate at which carbon-12 fuses into heavier elements" is incorrect because carbon-12 is stable and does not undergo fusion under normal environmental conditions. This technique works because living organisms maintain a roughly constant carbon-14 ratio with the atmosphere until death, after which the ratio decreases predictably.

Quantum tunneling occurs because a particle's wave function has a nonzero probability of extending beyond a classically forbidden energy barrier, allowing the particle to appear on the other side even without enough energy to go over it classically. The option "An electron jumping to a higher orbit after absorbing a photon" describes a simple energy absorption transition, not tunneling through a barrier. Tunneling is essential to phenomena like alpha decay and the operation of devices such as scanning tunneling microscopes.

Using \(L = L_0\sqrt{1-v^2/c^2} = 100\sqrt{1-0.36} = 100\sqrt{0.64} = 80\text{ m}\), the Earth observer measures a contracted length shorter than the proper length. The option "\(125\text{ m}\)" incorrectly applies the inverse relationship, expanding rather than contracting the length as observed from Earth. This calculation illustrates that length contraction only occurs along the direction of relative motion and only becomes significant at speeds that are a substantial fraction of \(c\).

Because an alpha particle carries away 2 protons and 2 neutrons, the parent nucleus loses 2 units of atomic number and 4 units of mass number, following the pattern \(^A_ZX \rightarrow {}^{A-4}_{Z-2}Y + \alpha\). The option "\(Z\) stays the same and \(A\) decreases by 4" is wrong because losing protons necessarily reduces the atomic number, not just the mass number. Tracking these changes correctly is essential for balancing nuclear decay equations on the exam.

Discrete spectral lines occur because electrons can only occupy specific quantized energy levels, and transitions between these fixed levels emit or absorb photons of specific, discrete frequencies. The option "The continuous nature of electron orbits" would instead predict a continuous spectrum, which contradicts the sharp, separated lines actually observed in experiments. This spectral evidence historically supported quantum models over classical continuous-orbit models of the atom.

From Earth's frame, the muon's internal decay clock runs slow due to time dilation, \(t = \gamma t_0\), extending its effective lifetime enough for many muons to survive the trip to sea level. The option "Muons travel faster than the speed of light" violates the fundamental postulate that no massive particle can reach or exceed \(c\). This classic experiment is one of the most direct empirical confirmations of relativistic time dilation.

Relativistic momentum \(p = \gamma m v\) includes the Lorentz factor, which diverges toward infinity as \(v\) approaches \(c\), meaning momentum grows much faster than the linear classical prediction near light speed. The option "The speed of light decreases relative to fast-moving objects" contradicts the core postulate that \(c\) is invariant for all observers. This divergence is precisely why an infinite amount of energy would be needed to push a massive object to exactly the speed of light.

The photon energy is \(E = hc/\lambda = 1240/200 \approx 6.2\text{ eV}\), and subtracting the work function using \(KE_{max} = E - \phi = 6.2 - 2.0 = 4.2\text{ eV}\) gives the maximum kinetic energy of the ejected electrons. The option "About \(6.2\text{ eV}\)" mistakenly reports the full photon energy without subtracting the energy needed to free the electron from the metal. This two-step calculation, finding photon energy then subtracting the work function, is the standard method for photoelectric effect problems.

Because the binding energy per nucleon curve peaks at iron-56, both fusing light nuclei together and splitting heavy nuclei apart move the resulting nuclei closer to this more stable, more tightly bound configuration, releasing the increase in binding energy as kinetic energy and radiation. The option "Fusion releases energy while fission actually absorbs energy" is factually wrong, since both processes are well known to release net energy under the right conditions. This curve explains why elements heavier than iron generally release energy through fission, while elements lighter than iron release energy through fusion.

From the platform observer's frame, the back of the train moves toward the point where the light was emitted while the front moves away, so the light reaches the back wall first even though both events are simultaneous in the train's own frame. The option "Both events occur simultaneously, exactly as observed on the train" ignores the relativity of simultaneity, which states that simultaneity depends on the observer's reference frame. This classic thought experiment is one of the clearest illustrations of why absolute simultaneity does not exist in special relativity.

Using \(N = N_0(1/2)^{t/T_{1/2}} = 80 \times (1/2)^{15/6} = 80 \times (1/2)^{2.5} \approx 80 \times 0.177 \approx 14.1\text{ g}\), the remaining mass reflects a non-integer number of half-lives. The option "About \(10\text{ g}\)" would correspond to exactly three half-lives, or 18 hours, which overshoots the given 15-hour interval. This calculation shows that exponential decay problems can be solved even when the elapsed time is not a whole-number multiple of the half-life.

The uncertainty principle, \(\Delta x \Delta p \gtrsim \hbar/2\), shows that shrinking the uncertainty in position \(\Delta x\) forces the uncertainty in momentum \(\Delta p\) to grow, meaning a confined electron must have a large minimum momentum and correspondingly large kinetic energy. The option "Confinement reduces the electron's mass" is physically incorrect, since an electron's rest mass is a fixed constant unaffected by spatial confinement. This reasoning is actually used to argue that electrons cannot exist inside atomic nuclei, since the required kinetic energy would be far greater than observed beta decay energies.

Applying \(E = mc^2 = 0.001 \times (3\times10^8)^2 = 0.001 \times 9\times10^{16} = 9\times10^{13}\text{ J}\) correctly accounts for squaring the speed of light before multiplying by the small converted mass. The option "About \(9\times10^{16}\text{ J}\)" mistakenly omits the mass factor, effectively calculating the energy for a full kilogram rather than one gram. This calculation demonstrates why even tiny amounts of mass converted in nuclear reactions release enormous quantities of energy compared to chemical reactions.

Moderators slow down the fast neutrons released by fission through repeated collisions, since slower thermal neutrons are much more likely to be captured by fissile nuclei like uranium-235 and trigger further fission events. The option "To absorb all neutrons and stop the chain reaction entirely" instead describes the role of control rods, which are a separate reactor component used to regulate or halt the reaction. Distinguishing moderators from control rods is important for understanding how reactors sustain a controlled, steady chain reaction.

Photon momentum is given by \(p = h/\lambda = (6.63\times10^{-34})/(500\times10^{-9}) \approx 1.33\times10^{-27}\text{ kg}\cdot\text{m/s}\), correctly dividing Planck's constant by the wavelength in meters. The option "About \(6.63\times10^{-34}\text{ kg}\cdot\text{m/s}\)" simply restates Planck's constant without dividing by the wavelength, ignoring the wavelength dependence entirely. This formula shows that even massless photons carry momentum, which explains phenomena like radiation pressure and the recoil observed in Compton scattering.

Alpha particles, being relatively massive and doubly charged, interact strongly and lose their energy quickly over a very short distance, while lighter, singly charged beta particles penetrate further, and uncharged, massless gamma photons interact only weakly and require dense material to absorb through multiple interactions. The option "Gamma rays are the least energetic form of radiation" is factually backward, since gamma photons are typically the most energetic of the three, not the least. Understanding this pattern of penetrating power is essential for choosing appropriate radiation shielding in medical and nuclear safety contexts.

The Pauli exclusion principle requires that no two electrons within the same atom share an identical set of four quantum numbers, which forces electrons to progressively fill higher orbitals and shells as atomic number increases, producing the repeating patterns seen across periods and groups of the periodic table. The option "All electrons in an atom must occupy the exact same energy level" directly contradicts the exclusion principle, since it is precisely what prevents electrons from all crowding into the lowest energy state. This principle, combined with quantized energy levels, underlies the entire structure of chemical bonding and atomic organization.

A GPS satellite's orbital speed causes its clock to run slightly slower due to special relativistic time dilation, while the weaker gravitational field at its altitude causes its clock to run slightly faster due to general relativistic effects, and since these effects are of different magnitudes rather than exactly canceling, both must be precisely calculated and corrected for accurate positioning. The option "The two effects exactly cancel" is incorrect because the general relativistic gravitational effect at GPS altitude actually outweighs the special relativistic speed effect, leaving a net timing difference that must be corrected. This real-world engineering application demonstrates that relativistic effects, though tiny, have measurable and important consequences in modern technology.