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1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
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6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
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9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

35. Special Relativity

Inertial Reference Frames

Physics

35. Special Relativity

Inertial Reference Frames

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Problem 43b

Textbook Question

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a ²³⁸U nucleus?

Verified step by step guidance

Step 1: Understand the problem. The question asks for the frequency of reflections of an alpha particle within a uranium-238 nucleus. This frequency is proportional to the speed of the alpha particle and the dimensions of the nucleus. We need to calculate the frequency using the formula for frequency in terms of speed and distance.

Step 2: Identify the relevant formula. The frequency of reflections can be calculated using the formula: $ f = \frac{v}{2d} $, where $ v $ is the speed of the alpha particle and $ d $ is the approximate diameter of the nucleus. The factor of 2 accounts for the round trip of the alpha particle across the nucleus.

Step 3: Estimate the diameter of the nucleus. The diameter of a uranium-238 nucleus can be approximated using the formula for nuclear radius: $ R = R_0 A^{1/3} $, where $ R_0 $ is the nuclear radius constant (approximately $ 1.2 \times 10^{-15} \, \text{m} $) and $ A $ is the mass number of the nucleus (238 for uranium-238). The diameter $ d $ is then $ 2R $.

Step 4: Determine the maximum speed of the alpha particle. The maximum speed can be estimated using the kinetic energy of the alpha particle, which is typically on the order of a few MeV. Use the relationship $ KE = \frac{1}{2}mv^2 $ to solve for $ v $, where $ m $ is the mass of the alpha particle (approximately $ 6.64 \times 10^{-27} \, \text{kg} $).

Step 5: Combine the values into the frequency formula. Substitute the calculated values of $ v $ and $ d $ into $ f = \frac{v}{2d} $ to find the frequency of reflections. Ensure units are consistent throughout the calculation (e.g., meters for distance, seconds for time).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons (essentially a helium nucleus). This process reduces the atomic number of the original nucleus by two and the mass number by four, leading to the formation of a new element. Understanding alpha decay is crucial for analyzing the stability of heavy nuclei and the mechanisms of nuclear reactions.

Nuclear Potential Well

The nuclear potential well is a model that describes the forces acting within a nucleus, where nucleons (protons and neutrons) are bound together by the strong nuclear force. The potential well creates a region where particles can exist, and the energy levels within this well determine the behavior of particles, including alpha particles. The concept is essential for understanding how alpha particles can be emitted from the nucleus and the conditions that affect their stability.

Reflection Frequency

Reflection frequency refers to the rate at which an alpha particle bounces off the boundaries of the nuclear potential well. This frequency is influenced by the particle's speed and the size of the nucleus. In the context of alpha decay, a higher reflection frequency increases the likelihood of the alpha particle escaping the nucleus, making it a key factor in determining the probability of decay events.

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Related Practice

Textbook Question

Consider the electron wave function $ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}$ where x is in cm. Draw a graph of |ψ(x)|² over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales.

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Textbook Question

A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

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Textbook Question

A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ mm where L = 2.0 mm. Determine the normalization constant c.

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Textbook Question

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. A ²³⁸U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a ²³⁸U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

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Textbook Question

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. What kind of photons are they—infrared, visible, or ultraviolet?

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Textbook Question

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. How long is the box in which the electrons are confined?

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Textbook Question

A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is briefly withdrawn, then reinserted, leaving the electron in the longer section of the box. What is the electron’s quantum state after the partition is back in place?

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Textbook Question

A finite potential well has depth U₀ = 2.00 eV. What is the penetration distance for an electron with energy (a) 0.50 eV, (b) 1.00 eV, and (c) 1.50 eV?

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