AP Unit 15 Problems

(a) Compare R and S by frequency, energy per photon, and vacuum momentum. Calculate both free-space speeds and both travel times, then evaluate the claim that the higher-energy photon reaches the detector first. State why the speed comparison does not depend on the chosen frequencies.

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(b) Explain which feature of the two-opening observations calls for wave behavior and which calls for particle-like energy transfer. A student says that the individual clicks rule out a wave description. Evaluate that statement for the whole set of observations.

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(c) A uniform slab on R’s path has index n = 1.50 in the supplied propagation model. At a stationary interface and with no absorption, compare R’s effective speed, frequency, wavelength, and energy inside the slab with its free-space values. Does that medium result change the free-space comparison in part a?

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(a) Select the statement that follows from the free-space photon model.

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(a) Derive the signed allowed circulation momentum pₛ(n) and the allowed energy Kₙ in terms of n, h, C, and m. Explain why a continuously adjustable classical ring momentum is incompatible with the stated wave boundary condition.

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(b) Calculate the allowed pₛ and K for n = 0, +1, −1, and +2. Compare the +1→+2 momentum step and energy gap. Test whether pₛ = 1.5h/C is allowed, and whether changing +1 to −1 must change the energy.

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(c) Sketch the allowed energy levels for |n| = 0, 1, 2, and 3 on a labeled energy axis using the calculated K₁. Compare the successive gaps, and evaluate: “This proves every Cartesian momentum component of every bound quantum system has discrete measured values.” State the defensible conclusion from this model.

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(a) Draw a labeled spatial schematic of the neutral atom showing the signs of charge, the small nuclear region, and the electron outside it. Calculate the nuclear-to-atomic radius ratio and explain why a readable nucleus inset is not to scale. Evaluate a diagram that places negative electrons inside the positive nucleus while retaining the same particle counts.

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(b) For the separate Bohr model, draw the instantaneous velocity, electric force, and acceleration directions at x = +r. Calculate the force magnitude and speed by equating electric attraction to centripetal force. Explain why the force is not along the velocity.

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(c) Calculate the electron’s de Broglie wavelength and the approximate number of wavelengths around this Bohr circumference. Explain why force balance alone does not select allowed energy states, and give the historical model’s useful scope.

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(a) Identify the best supplied reference for source A and for gas B, using all measured line positions and their uncertainty. Explain why the two setups can identify different materials and why a single coincident line would be weaker evidence.

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(b) Draw aligned qualitative emission and absorption spectra with a common wavelength axis from 380 to 740 nm. Mark the measured line positions, distinguish bright lines from dark lines on a continuous background, and explain the atomic energy-transfer direction in each setup.

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(c) A separate hypothetical single-electron atom has levels −8.0, −3.0, and −1.0 eV. A photon corresponding to the −3.0→−1.0 eV gap passes through gas whose atoms are all in the −8.0 eV ground state. Must that line be absorbed? Explain using its energy, the occupied starting level, and the ground-state ionization threshold. Compare the removal energies from all three levels and explain why the ground state requires the greatest removal energy.

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(a) Explain which interaction dominates nucleon interactions at typical nuclear binding separations and why a nucleus containing mutually repelling protons can be bound. Include proton–neutron and neutron–neutron interactions. Calculate the proton–proton electric-force magnitude at 1.5 fm, draw its direction, and explain why the short-range binding interaction does not dominate two protons separated by 10⁻¹⁰ m.

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(b) For the supplied fusion channel, verify total nucleon number and electric charge, compute Q, and explain which energy is conserved and which mass quantity can change. Name the allowed forms of released energy in nuclear processes, then use the stated no-photon channel to account for this Q.

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(c) Draw the final-product momentum vectors in the center-of-momentum frame, state their magnitude and direction relationship, and derive the kinetic-energy ratio Kn/KHe. Calculate both kinetic energies and check their sum. Evaluate the claim that two products with equal momentum magnitudes must receive equal kinetic energies.

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(d) Contrast the supplied fusion and fission channels by what the nuclei do. Verify the fission channel’s total nucleon number and charge, including every free neutron. Explain why a positive-Q fission pathway can still need neutron capture or excitation rather than making every parent split immediately; relate fusion’s close-approach requirement to the short range discussed in part a.

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(a) Select the explanation that accounts for both the nucleon types and the relevant distance scale.

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