Problem 55 - Fundamentals of Physics

Question

A certain particle of mass \(m\) has momentum of magnitude \(m c .\) What are (a) \(\beta,(b) \gamma,\) and \((c)\) the ratio \(K / E_{0} ?\)

Step-by-Step Solution

Answer

(a) \(\beta = 1\), (b) \(\gamma\) is infinite, (c) \(K / E_0\) is infinite.

Step 1: Understanding Momentum and Velocity

Given the problem, the momentum of a particle is defined as \(p = mv\) where \(v\) is velocity. Here, it's given that \(p = mc\), meaning the particle's velocity is equal to the speed of light \(c\), which would make the relativistic momentum equation appropriate since particles can't exceed the speed of light.

Step 2: Relativistic Factor \(\beta\)

\(\beta\) is defined as the ratio of the particle's velocity \(v\) to the speed of light \(c\). Here, since \(p = mc\), we interpret this as a relativistic scenario. Hence, \(\beta = \frac{v}{c} = \frac{c}{c} = 1.\)

Step 3: Determine Relativistic Gamma \(\gamma\)

The relativistic factor \(\gamma\) is given by \(\gamma = \frac{1}{\sqrt{1 - \beta^2}}\). Since \(\beta = 1\), this implies an infinite value for \(\gamma\), indicating that such a scenario is more theoretical because \(\beta\) approaches 1 but cannot equal it exactly.

Step 4: Calculate Ratio \(\frac{K}{E_0}\)

The relativistic kinetic energy is given by \(K = (\gamma - 1)mc^2\) and rest energy is \(E_0 = mc^2\). Hence, the ratio \(\frac{K}{E_0} = \frac{\gamma - 1}{1}\). With \(\gamma\) approaching infinity as \(\beta\) approaches 1, this ratio essentially becomes infinite.

Key Concepts

Momentum

Momentum is a fundamental concept in physics, but it takes on a special form in relativistic mechanics. Generally, momentum (\( p \)) is the product of mass (\( m \)) and velocity (\( v \)). In the realm of relativistic mechanics, however, momentum is influenced by the relativistic factor gamma (\( \gamma \)), which alters its traditional expression.
In relativistic terms, the momentum equation becomes \( p = \gamma mv \), accounting for effects as speeds approach the speed of light (\( c \)). When the particle's momentum is given as \( mc \), it suggests that the particle is moving at a velocity comparable to the speed of light.
This increased momentum compared to classical mechanics is what accounts for shifts like increased mass effects as speeds near light speed. Understanding momentum in this context emphasizes how the classical mechanics' assumptions change as velocities increase significantly.

Relativistic Factor Gamma

The relativistic factor, commonly denoted as \( \gamma \), plays a critical role when dealing with relativistic speeds. It helps explain how time, mass, and other physical quantities transform when approaching the speed of light.
\( \gamma \) is defined by the equation \( \gamma = \frac{1}{\sqrt{1 - \beta^2}} \), where \( \beta = \frac{v}{c} \). As the particle velocity \( v \) approaches the speed of light \( c \), \( \beta \) nears 1, causing \( \gamma \) to increase steadily.

• For small velocities compared to the speed of light, \( \gamma \) is close to 1, similar to classical mechanics.
• As \( v \) approaches \( c \), \( \gamma \) escalates towards infinity, indicating significant relativistic effects.

This increase affects momentum, energy, and time, highlighting their interconnected nature at relativistic scales.

Kinetic Energy

Kinetic energy in relativistic contexts differs from its classical counterpart. In classical mechanics, kinetic energy (\( K \)) is given by \( \frac{1}{2}mv^2 \). However, in relativistic mechanics, things change.
The relativistic kinetic energy is expressed as \( K = (\gamma - 1)mc^2 \). This formula considers the increase in energy due to relativistic effects as velocity approaches the speed of light.

• Relativistic kinetic energy accounts for the dramatic changes in mass and energy relations at high speeds.
• As the relativistic factor \( \gamma \) increases, so does kinetic energy, leading to significant deviations from classical predictions.

The ratio of kinetic energy to rest energy (\( \frac{K}{E_0} \)) becomes a crucial point, revealing the proportional increase in energy required to continue accelerating a particle as it nears light speed.

Speed of Light

The speed of light, denoted as \( c \), is a fundamental constant in physics, precisely measured as approximately 299,792,458 meters per second. It represents the ultimate speed at which energy, matter, and information can travel, serving as a critical limit in relativistic physics.
Because nothing with mass can reach or exceed this limit, the speed of light plays a fundamental role in the equations and principles of relativistic mechanics.

• In the context of momentum, the speed of light ensures that no massive object can reach velocities equal or greater than \( c \).
• It is used as a factor \( \beta \) in comparing particle velocities to the speed of light, crucial in determining relativistic effects.
• The speed of light also grounds the equations used for determining relativistic energy and time dilation, making it a cornerstone of modern physics.

Understanding its role provides insight into how modern physics distinguishes itself from classical theories, encapsulating the changes in dynamics at extreme velocities.