91Ó°ÊÓ

A certain radioactive (parent) nucleus transforms to a different (daughter) nucleus by emitting an electron and a neutrino. The parent nucleus was at rest at the origin of an \(x y\) coordinate system. The electron moves away from the origin with linear momentum \(\left(-1.2 \times 10^{-22} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\right) \hat{\mathrm{i}} ;\) the neutrino moves away from the origin with linear momentum \(\left(-6.4 \times 10^{-23} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\right) \hat{\mathrm{j}}\). What are the (a) magnitude and (b) direction of the linear momentum of the daughter nucleus? (c) If the daughter nucleus has a mass of \(5.8 \times\) \(10^{-26} \mathrm{~kg}\), what is its kinetic energy?

Step by step solution

01

Apply Conservation of Momentum

The law of conservation of momentum states that the total momentum before the decay is equal to the total momentum after decay. Since the parent nucleus was initially at rest, its initial momentum is zero. So, the sum of the momenta of the electron, neutrino, and daughter nucleus should be zero. This can be mathematically represented as \( \vec{p}_{\text{electron}} + \vec{p}_{\text{neutrino}} + \vec{p}_{\text{daughter}} = \vec{0} \).

02

Calculate Daughter Nucleus Momentum Components

Given the momentum of the electron is \( \vec{p}_{\text{electron}} = (-1.2 \times 10^{-22}) \hat{i} \) kg·m/s, and the momentum of the neutrino is \( \vec{p}_{\text{neutrino}} = (-6.4 \times 10^{-23}) \hat{j} \) kg·m/s. Using the conservation of momentum:\[ \vec{p}_{\text{daughter}} = - \vec{p}_{\text{electron}} - \vec{p}_{\text{neutrino}} \]\[ \vec{p}_{\text{daughter}} = (1.2 \times 10^{-22}) \hat{i} + (6.4 \times 10^{-23}) \hat{j} \] kg·m/s.

03

Calculate Magnitude of Daughter Nucleus Momentum

The magnitude of the momentum vector \( \vec{p}_{\text{daughter}} = (1.2 \times 10^{-22}) \hat{i} + (6.4 \times 10^{-23}) \hat{j} \) can be found using the Pythagorean theorem:\[ |\vec{p}_{\text{daughter}}| = \sqrt{(1.2 \times 10^{-22})^2 + (6.4 \times 10^{-23})^2} \]\[ = \sqrt{1.44 \times 10^{-44} + 4.096 \times 10^{-45}} \]\[ \approx 1.36 \times 10^{-22} \text{ kg·m/s} \].

04

Determine the Direction of the Momentum

The direction of the momentum vector can be found using the tangent function:\[ \theta = \tan^{-1}\left(\frac{p_j}{p_i}\right) = \tan^{-1}\left(\frac{6.4 \times 10^{-23}}{1.2 \times 10^{-22}}\right) \]\[ \theta \approx \tan^{-1}(0.533) \approx 28.1^\circ \], directed counterclockwise from the positive \(x\)-axis.

05

Calculate Kinetic Energy of Daughter Nucleus

The kinetic energy (KE) of the daughter nucleus can be calculated using the formula:\[ KE = \frac{1}{2}mv^2 \]Where \( v \) is the velocity, derived from momentum \( p = mv \):\[ v = \frac{|\vec{p}_{\text{daughter}}|}{m} = \frac{1.36 \times 10^{-22}}{5.8 \times 10^{-26}} \approx 2.34 \times 10^{3} \text{ m/s} \]Substitute this velocity back into the kinetic energy formula:\[ KE = \frac{1}{2} \times 5.8 \times 10^{-26} \times (2.34 \times 10^{3})^2 \]\[ KE \approx 1.59 \times 10^{-19} \text{ J} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radioactive Decay

Radioactive decay involves the transformation of an unstable atomic nucleus into a more stable one by releasing particles. This process often emits high-energy particles and radiation, which we call ionizing radiation. In this case, a parent nucleus decays by emitting an electron and a neutrino.

Here's what's important to understand about this transformation:

• Parent and Daughter Nuclei: The unstable nucleus (parent) transforms into a new nucleus (daughter) after the decay.
• Emission Particles: The emitted electron is a beta particle. Neutrinos, almost massless particles that carry away energy, accompany this emission.
• Process Occurrence: Such decays occur naturally and are part of the processes that power stars and explosions in space.

The decay causes changes in the atomic structure, resulting in the release and redistribution of energy and momentum, central concepts when solving related physics problems.

Linear Momentum

Linear momentum is a vector quantity defined as the product of an object's mass and velocity. It is conserved in isolated systems, meaning the total momentum before an event like radioactive decay is equal to the total momentum after. This principle helps answer questions about particle interactions.

In the provided exercise:

• Initial Momentum: The parent nucleus is initially at rest, so its total momentum is zero.
• Conservation Law: The sum of the linear momentum vectors of the electron, neutrino, and daughter nucleus must also equal zero post-decay.
• Calculating Momentum: The momentum components for the daughter nucleus can be calculated using vector addition, ensuring both horizontal and vertical components consider the respective contributions of the emitted particles.

Understanding these components allows us to derive the final velocity and kinetic energy, crucial for deeper analysis of decay effects.

Kinetic Energy

Kinetic energy in physics refers to the energy an object possesses due to its motion. Calculating the kinetic energy of a moving object involves knowing its mass and velocity. Kinetic energy is given by the formula:\[ KE = \frac{1}{2} mv^2 \]In this context:

• Mass and Velocity: For the daughter nucleus, its mass is known, and its velocity can be derived from the linear momentum.
• Energy Conversion: As momentum provides insight into the object's movement, inserting its velocity into the kinetic energy formula reveals the energy associated with this motion.
• Solving for KE: After determining linear momentum and velocity, substitute these values back into the kinetic energy equation to find the result.

This comprehension of kinetic energy helps in quantifying the dynamic aspects of particle decay, essential for practical applications in physics.

Physics Problem Solving

Physics problem solving often involves breaking down complex processes into simpler parts, like using fundamental principles such as conservation laws. Systematic approaches and employing mathematical tools are key to efficiently tackling problems.

In the discussed exercise:

• Step-by-Step Analysis: Start with broad principles like conservation, then crack down into specific vectors and equations.
• Mathematical Representation: Use mathematical expressions to solve for unknowns like momentum components and kinetic energy.
• Verification: Double-check calculations and ensure all steps align with physical laws, providing accurate answers consistent with reality.

Approaching physics problems this way nurtures critical thinking, promotes accuracy, and enhances understanding of physical phenomena in various contexts, essential for mastering physics concepts.