Problem 51 - Fundamentals of Physics

Question

What must be the momentum of a particle with mass \(m\) so that the total energy of the particle is \(3.00\) times its rest energy?

Step-by-Step Solution

Answer

The particle's momentum is \(p = 2mc\sqrt{2}\).

Step 1: Understanding the Problem

The problem asks us to find the momentum of a particle whose total energy is 3 times its rest energy. We start by noting that the rest energy of a particle with mass \(m\) is \(E_0 = mc^2\), where \(c\) is the speed of light.

Step 2: Total Energy Expression

The total energy \(E\) of a particle with momentum \(p\) and mass \(m\) is given by the formula \(E = \sqrt{(pc)^2 + (mc^2)^2}\). We set this equal to 3 times the rest energy: \(E = 3E_0 = 3mc^2\).

Step 3: Solve for Momentum

Plug the total energy condition into the energy equation: \(3mc^2 = \sqrt{(pc)^2 + (mc^2)^2}\). Square both sides to eliminate the square root, resulting in \((3mc^2)^2 = (pc)^2 + (mc^2)^2\).

Step 4: Simplify the Equation

Simplify the equation: \(9m^2c^4 = p^2c^2 + m^2c^4\). By subtracting \(m^2c^4\) from both sides, we have \(8m^2c^4 = p^2c^2\).

Step 5: Isolate Momentum

Divide both sides by \(c^2\) to isolate \(p^2\): \(p^2 = 8m^2c^2\). Take the square root of both sides to solve for \(p\): \(p = mc\sqrt{8}\).

Step 6: Final Expression for Momentum

Simplifying further, \(p = mc\sqrt{8} = mc \times 2\sqrt{2}\). Hence, the momentum of the particle is given by \(p = 2mc\sqrt{2}\).

Key Concepts

Rest Energy

Rest energy is one of the fundamental concepts in understanding the behavior of particles. It refers to the energy that is held within a particle when it is not in motion, relative to an observer. This energy is given by the formula \(E_0 = mc^2\), where \(m\) is the rest mass of the particle, and \(c\) is the speed of light, which is a constant.

This concept shows the relationship between mass and energy, suggesting that even stationary particles possess inherent energy due to their mass.

Key aspects of rest energy include:

• It is independent of the particle's velocity.
• The term highlights the intrinsic energy due to mass alone.
• It provides a reference point for calculating changes in a particle's energy.

Rest energy is crucial in relativistic physics, forming a baseline to compare against the total energy when objects are in motion.

Total Energy

The total energy of a particle conveys much more information than just its rest energy. In relativistic physics, the total energy \(E\) is the sum of the particle's rest energy and kinetic energy, accounting for motion.

This is represented by the equation: \(E = \sqrt{(pc)^2 + (mc^2)^2}\), where \(p\) is the relativistic momentum of the particle, and \(c\) is the speed of light.

Key points to understand total energy include:

• It comprises both rest energy and the energy due to motion.
• It indicates how much energy would be released if a particle were to come to a stop.
• The formula integrates both kinetic and rest energy, emphasizing their coexistence at relativistic speeds.

In many problems, such as the exercise here, we see how total energy compares to multiples of rest energy, offering insights into momentum under relativistic conditions.

Relativistic Energy Equation

The relativistic energy equation interlinks the concepts of rest energy, kinetic energy, and total energy. It is vital in understanding the energy dynamics when particles travel near the speed of light.

The equation \(E = \sqrt{(pc)^2 + (mc^2)^2}\) forms the cornerstone of relativistic energy calculations. Here's how this equation serves different purposes:

• It provides a way to calculate total energy, combining momentum and rest energy.
• It explains energy conservation across various reference frames.
• It is used in fields like particle physics to determine energies during collisions and accelerations.

The relativistic energy equation goes beyond classical mechanics by recognizing that mass and energy are interconvertible at high speeds, an insight that reshaped modern physics. This ability to calculate the energy considering both rest mass and momentum allows us to predict the behavior of particles accurately in extreme conditions.