Problem 83 - Fundamentals of Physics

Question

What are (a) \(K,(\) b) \(E\), and \((\mathrm{c}) p(\) in \(\mathrm{GeV} / c)\) for a proton moving at speed \(0.990 c\) ? What are (d) \(K\), (e) \(E\), and (f) \(p\) (in \(\mathrm{MeV} / c\) ) for an electron moving at speed \(0.990 c ?\)

Step-by-Step Solution

Answer

For the proton: (a) \(K = 6.064\text{ GeV}\), (b) \(E = 6.646\text{ GeV}\), (c) \(p = 6.640\text{ GeV/c}\). For the electron: (d) \(K = 3.797\text{ MeV}\), (e) \(E = 3.808\text{ MeV}\), (f) \(p = 3.796\text{ MeV/c}\).

Step 1: Identify the Known Values

For a proton, mass is \( 938 \) MeV/c², and for an electron, mass is \( 0.511 \) MeV/c². In both scenarios, speed \( v = 0.990c \).

Step 2: Calculate Lorentz Factor \( \gamma \)

The Lorentz factor \( \gamma \) is given by:\[ \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]Substituting \( v = 0.990c \):\[ \gamma = \frac{1}{\sqrt{1 - (0.990)^2}} \approx 7.088 \]

Step 3: Calculate Kinetic Energy \( K \) for Proton and Electron

Kinetic energy for relativistic speeds is \( K = (\gamma - 1)mc^2 \).For the proton:\[ K_p = (7.088 - 1) \times 938 \text{ MeV} \approx 6064 \text{ MeV} = 6.064 \text{ GeV} \]For the electron:\[ K_e = (7.088 - 1) \times 0.511 \text{ MeV} \approx 3.797 \text{ MeV} \]

Step 4: Calculate Total Energy \( E \) for Proton and Electron

The total energy is \( E = \gamma mc^2 \).For the proton:\[ E_p = 7.088 \times 938 \text{ MeV} \approx 6646 \text{ MeV} = 6.646 \text{ GeV} \]For the electron:\[ E_e = 7.088 \times 0.511 \text{ MeV} \approx 3.808 \text{ MeV} \]

Step 5: Calculate Momentum \( p \) for Proton and Electron

The momentum is \( p = \gamma mv \).Since \( p \) is often expressed per \( c \), use \( p = \gamma mv/c \). For the proton:\[ p_p = 7.088 \times 938 \text{ MeV/c}^2 \times 0.990c / c \approx 6640 \text{ MeV/c} = 6.640 \text{ GeV/c} \]For the electron:\[ p_e = 7.088 \times 0.511 \text{ MeV/c}^2 \times 0.990c / c \approx 3.796 \text{ MeV/c} \]

Key Concepts

Understanding the Lorentz Factor

The Lorentz factor (\( \gamma \)) is crucial in relativity when objects approach the speed of light (c). It relates the observed length, time, and energy from different inertial frames.

• Formula: \[ \gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2 }} \]
• Importance: As speed increases towards the speed of light, the Lorentz factor becomes significantly greater than one.
• At high speeds, it affects how we measure time (time dilation), length (length contraction), and energy.

In the exercise, for a speed of 0.990c, the Lorentz factor is approximately 7.088, showing how much energy, momentum, and kinetic calculations differ from non-relativistic speeds.

Exploring Kinetic Energy at Relativistic Speeds

Kinetic energy (\(K\)) at relativistic speeds diverges from classical mechanics. In classical physics, \(K\ = \frac{1}{2}mv^2\). However, when objects move close to light speed, we use:\[ K = (\gamma - 1) mc^2 \]Key points:

• Kinetic energy increases substantially at relativistic speeds.
• This equation accounts for energy increases due to inertia growth at high speeds.

For the proton in our example, its kinetic energy is about 6.064 GeV, while the electron has approximately 3.797 MeV. Understanding how kinetic energy changes is vital when analyzing high-speed particle behavior.

Calculating Total Energy for Fast-Moving Particles

The total energy (\(E\)) of an object is the sum of its rest mass energy and its kinetic energy. When dealing with relativistic speeds, it's crucial to use:\[ E = \gamma mc^2 \]

• Total energy grows with speed due to increases in both mass and kinetic energy.
• In the exercise, for the proton, we calculate total energy to be around 6.646 GeV.
• For the electron, total energy is about 3.808 MeV.

This concept is essential when dealing with high-speed particles like those found in particle accelerators, where energies significantly exceed rest mass energies.

The Role of Momentum in Relativity

Momentum (\(p\)) in relativity differs significantly from its classical counterpart. In relativistic contexts, it's given by:\[ p = \gamma mv \]

• The factor \(\gamma\) accounts for velocity increases near the speed of light.
• This equation shows momentum is not just dependent on speed, but also on relativistic mass.

For example, the proton in our exercise achieves a momentum of about 6.640 GeV/c, while the electron has roughly 3.796 MeV/c. Appreciating the adjustments in momentum at relativistic speeds is key for understanding particle motion.

Why Focus on the Proton?

Protons are subatomic particles found in atomic nuclei. They are often used in particle physics due to their charge and relatively large mass:

• Mass: 938 MeV/c²
• Application: Protons are a primary subject in particle collisions and high-energy physics experiments.

In our scenario, the proton at 0.990c offers a glimpse into how relativistic speeds alter energy and momentum compared to particles at rest or moving slowly. With substantial mass, protons reveal significant insights into energy transformations and interactions at high speeds.

Understanding Electron Speed

Electrons, while much lighter than protons with a mass of 0.511 MeV/c², play a critical role in atomic and subatomic physics. Their speed can greatly influence calculations at high velocities:

• Mass: 0.511 MeV/c²
• Behavior: Lightness means that relativistic effects, such as kinetic energy, are proportionally less than for protons.

Despite the smaller energy values compared to protons, at 0.990c, even electrons display significant relativistic adjustments, highlighting the importance of relativity at all mass scales.

What Are Relativistic Speeds?

Relativistic speeds refer to velocities close to the speed of light. At these speeds, classical Newtonian mechanics is no longer sufficient to describe motion:

• Effects: Objects appear heavier and time slows down, a phenomenon explained by Einstein's theory of relativity.
• The exercise indicates: With both protons and electrons moving at 0.990c, relativistic calculations are necessary.

Understanding and calculating the physics of relativistic speeds are essential in fields like astrophysics and high-energy particle physics, where objects frequently travel close to light speed.