Question

Find the momentum of a particle of mass \(m\) in order that its total energy be three times its rest energy.

Short answer

The momentum of the particle is given by \(p = mc \sqrt{8}\).

Step-by-step solution

Write down the given information

The total energy \(E\) is three times the rest energy, so we have \(E = 3mc^2\).

Use the energy-momentum relation

The equation for the energy-momentum relation is \(E^2 = (pc)^2 + (mc^2)^2\). We are looking for \(p\).

Substitute the total energy into the energy-momentum relation

Substitute \(E = 3mc^2\) into the energy-momentum relation. This gives us \((3mc^2)^2 = (pc)^2 + (mc^2)^2\).

Solve for momentum

After solving for \(p\), the momentum, you get \(p = mc \sqrt{8}\).

Key concepts

Special Relativity

At the turn of the 20th century, physics underwent a revolution with the formulation of special relativity by Albert Einstein. Central to special relativity are two postulates: the laws of physics are the same for all non-accelerating observers, and the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer.

One of the groundbreaking consequences of these postulates is the idea that mass and energy are equivalent, summarized in the famous equation, \(E = mc^2\), where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light. This intertwines with concepts such as time dilation and length contraction, radically changing our understanding of space and time. Notably, special relativity has a deep impact on understanding the energy and momentum of particles, as illustrated in the exercise we explore.

Particle Physics

Particle physics is the branch of physics that studies the elementary constituents of matter and their interactions. Within this field, scientists search for the fundamental particles from which everything else is composed—quarks, leptons, and bosons—and how they combine to form protons, neutrons, atoms, and so on.

In particle physics, the properties of these particles, such as mass, charge, and spin, are studied under various conditions. The behavior of particles is often counterintuitive and cannot be explained by classical physics; hence, concepts from special relativity and quantum mechanics are essential. The exercise presented ties directly into particle physics by examining the momentum of a particle, a key attribute in determining how particles move and interact in high-energy environments.

Momentum of a Particle

Momentum is a fundamental concept in physics, representing the quantity of motion of a moving body and is directly tied to the mass and velocity of that body. In classical physics, the momentum \(p\) of a particle is given by the product of its mass \(m\) and velocity \(v\), typically written as \(p = mv\).

In the realm of high-speed particles and special relativity, however, the classical definition is not adequate, as one has to consider the relativistic effects on mass and velocity. The relation \(p = mv\) gets modified to account for these effects, which can include an increase in mass with velocity. The problem at hand uses the relativistic expression for momentum, which involves finding a relationship between the particle's energy and momentum different from the classical Newtonian physics.

Rest Energy

Rest energy refers to the intrinsic energy of a particle when it is not in motion. According to special relativity, even a stationary particle has energy due to its mass, which is described by the equation \(E_0 = mc^2\), where \(E_0\) stands for rest energy. This reveals a profound insight: mass can be considered a form of energy.

The exercise we're examining deals with a particle whose total energy is a multiple of its rest energy. This is a common situation in particle physics where particles are accelerated to high speeds, and their total energy (kinetic plus rest energy) becomes a significant multiple of their rest energy. By solving the problem and finding the momentum for a particle with total energy three times its rest energy, one gains a deeper comprehension of the interplay between energy and momentum in the relativistic regime.