Question

(a) What is a particle's speed if its kinetic energy \(T\) is equal to its rest energy? ( \(\mathbf{b}\) ) What if its energy \(E\) is equal to \(n\) times its rest energy?

Short answer

For (a), the speed is \( \frac{\sqrt{3}}{2}c \). For (b), the speed is \( c\sqrt{1 - \frac{1}{n^2}} \).

Step-by-step solution

Understanding the Rest Energy

First, let's recognize that the rest energy, denoted as \( E_0 \), of a particle is given by the equation \( E_0 = mc^2 \), where \( m \) is the rest mass of the particle and \( c \) is the speed of light.

Kinetic Energy Equals Rest Energy

In this scenario, we know that the kinetic energy \( T \) of the particle is equal to its rest energy. Since kinetic energy in relativity is \( T = \, \gamma mc^2 - mc^2 \), set \( T = mc^2 \). This implies \( \gamma mc^2 - mc^2 = mc^2 \).

Solving for Lorentz Factor \( \gamma \)

Simplify the equation from the previous step: \( \gamma mc^2 = 2mc^2 \). Cancel \( mc^2 \) terms on both sides to find \( \gamma = 2 \).

Calculating Speed from Lorentz Factor

The Lorentz factor is defined as \( \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \). Substitute \( \gamma = 2 \) and solve for \( v \): \( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 2 \). This implies \( \sqrt{1-\frac{v^2}{c^2}} = \frac{1}{2} \).

Final Calculation of Speed

Square both sides: \( 1 - \frac{v^2}{c^2} = \frac{1}{4} \). Solve for \( \frac{v^2}{c^2} \): \( \frac{v^2}{c^2} = \frac{3}{4} \), so \( v = c\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}c \).

Understanding Energy Equals \( n \) Times Rest Energy

In this scenario, the total energy \( E \) is given as \( E = nE_0 = nmc^2 \). Since the total energy \( E = \gamma mc^2 \), equating gives \( \gamma mc^2 = nmc^2 \), which simplifies to \( \gamma = n \).

Calculating Speed for \( n \times \text{Rest Energy} \)

Substitute \( \gamma = n \) into the Lorentz factor equation: \( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = n \). Solving gives \( \sqrt{1-\frac{v^2}{c^2}} = \frac{1}{n} \).

Final Speed Calculation for \( n \times \text{Rest Energy} \)

Square both sides to get \( 1 - \frac{v^2}{c^2} = \frac{1}{n^2} \), solving for \( \frac{v^2}{c^2} \) gives \( \frac{v^2}{c^2} = 1 - \frac{1}{n^2} \), so \( v = c\sqrt{1 - \frac{1}{n^2}} \).

Key concepts

Lorentz Factor

The Lorentz Factor, symbolized by the Greek letter \( \gamma \), is a crucial concept in relativistic physics. It plays a central role when objects travel at a significant fraction of the speed of light, \( c \). The Lorentz factor is defined by the equation \( \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \), where \( v \) is the velocity of the object.

This factor accounts for time dilation and length contraction, which are important effects predicted by Einstein's theory of relativity. For example, as an object's speed approaches the speed of light, its Lorentz factor increases significantly. At the speed where kinetic energy matches rest energy, the Lorentz factor is 2, highlighting that relativistic effects are involved.

Understanding the Lorentz factor helps explain why particles in accelerators require a lot of energy to increase their speed further, as their effective mass grows due to relativity, making it harder to accelerate them.

Kinetic Energy

In relativistic terms, kinetic energy is not as straightforward as in classical physics. It's defined by the equation \( T = \gamma mc^2 - mc^2 \), where \( m \) is the rest mass of the particle and \( \gamma \) is the Lorentz factor. This shows that kinetic energy grows significantly as speeds increase and approach the speed of light.

• Classically, kinetic energy is \( T = \frac{1}{2}mv^2 \), but this formula breaks down at high velocities.
• Relativistically, an increase in velocity results in a disproportionate rise in kinetic energy.

When the particle's kinetic energy equals its rest energy (\( T = mc^2 \)), this dramatic change is due to the exponential nature of the Lorentz factor.

Rest Energy

Rest energy relates to the energy inherent in a mass when it's not in motion. Deriven from Einstein's famous equation \( E_0 = mc^2 \), rest energy reveals an intrinsic energy within all matter, linked inseparably to its mass.

• This concept implies that mass can be converted into energy, as shown in nuclear fission or fusion processes.
• When kinetic energy of a particle equals its rest energy, it indicates significant relativistic speeds.

Understanding rest energy is crucial because it represents the baseline energy of any particle, against which other forms of energy are compared. In the problems involving relativistic kinetics, the rest energy is often a starting point for calculations of total energy in systems.

Speed of Light

The speed of light, denoted by \( c \), is more than just a constant; it's a fundamental limit in the universe. It represents the maximum speed at which information and matter can travel. Specifically, \( c \approx 3 \times 10^8 \) meters per second in a vacuum.

In the realm of relativistic physics, the speed of light also serves as a benchmark against which all other speeds are measured.

• At speeds close to \( c \), time and space behave differently due to relativistic effects.
• Rest energy itself is tied to \( c \) with its equation \( E_0 = mc^2 \).

Moreover, approaching this speed magnifies the relativistic effects described by the Lorentz factor, making it a critical aspect of modern physics explorations. Understanding the speed of light's unique role illuminates why certain calculations, like those in kinetic energy and Lorentz factor, vary so distinctly from classical predictions.