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Chapter 27: Problem 34

A particle has a rest mass of \(6.64 \times 10^{-27} \mathrm{~kg}\) and a momentum of \(2.10 \times 10^{-18} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). (a) What is the total energy (kinetic plus rest energy) of the particle? (b) What is the kinetic energy of the particle? (c) What is the ratio of the kinetic energy to the rest energy of the particle?

Short Answer

Expert verified
Total energy is calculated using relativistic energy formula. Kinetic energy is the total energy minus rest energy. The ratio is kinetic energy divided by rest energy.

Step by step solution

01

Understand Rest Mass and Energy Formulas

To solve this problem, we need to find the total energy and kinetic energy of the particle using relativistic formulas. The total energy \(E\) of a particle is given by \(E = \sqrt{(p c)^2 + (m_0 c^2)^2}\), where \(p\) is the momentum, \(c\) is the speed of light, and \(m_0\) is the rest mass.
02

Calculate Total Energy

Plug in the given values into the total energy formula. Here, \(p = 2.10 \times 10^{-18} \text{ kg} \cdot \text{m/s}\), \(m_0 = 6.64 \times 10^{-27} \text{ kg}\), and \(c = 3.00 \times 10^8 \text{ m/s}\).\[E = \sqrt{(2.10 \times 10^{-18} \cdot 3.00 \times 10^8)^2 + (6.64 \times 10^{-27} \cdot 3.00 \times 10^8)^2}\]Calculate \(p c\) and \(m_0 c^2\), then find \(E\).
03

Calculate Rest Energy

Calculate the rest energy \(E_0\) of the particle using the formula \(E_0 = m_0 c^2\).Substitute \(m_0 = 6.64 \times 10^{-27} \text{ kg}\) and \(c = 3.00 \times 10^8 \text{ m/s}\).\[E_0 = 6.64 \times 10^{-27} \times (3.00 \times 10^8)^2\]
04

Extract Kinetic Energy

The kinetic energy \(K\) is given by the difference between the total energy \(E\) and the rest energy \(E_0\), or \(K = E - E_0\). Use the values calculated in the previous steps to find \(K\).
05

Calculate the Ratio of Kinetic Energy to Rest Energy

Calculate the ratio \(\frac{K}{E_0}\) using the kinetic energy \(K\) and the rest energy \(E_0\).This will give you an understanding of how much more the kinetic energy is compared to the rest energy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rest Mass
Rest mass, often denoted as \(m_0\), is the mass of an object when it is at rest relative to an observer. It is a fundamental concept in both classical and relativistic physics. Rest mass does not change regardless of the object's speed or location. In our example, the rest mass of the particle is \(6.64 \times 10^{-27} \text{ kg}\). This value is used in calculations involving relativistic energy equations.
  • Rest mass is invariant and stays constant even when the particle is in motion.
  • In the realm of relativistic physics, rest mass is crucial for understanding other energy concepts.
Rest mass forms the basis for calculating rest energy and total energy, making it a vital component of relativistic equations.
Momentum
Momentum is the quantity of motion an object possesses. In relativistic physics, momentum \(p\) is affected by the high speeds at which particles travel. The formula for momentum in this context often includes the speed of light \(c\) due to its relationship with energy. For the given particle, the momentum is \(2.10 \times 10^{-18} \text{ kg} \cdot \text{m/s}\).
  • Momentum combines mass and velocity; it's represented by \(p = m \cdot v\) in classical mechanics.
  • Here, it is necessary to consider relativistic effects, requiring adjustments to our calculations.
The momentum plays a critical role in determining the total relativistic energy of the particle.
Kinetic Energy
Kinetic energy is the energy that a particle possesses due to its motion. In relativity, the kinetic energy \(K\) differs from classical calculations. It's computed as the difference between the total energy \(E\) and the rest energy \(E_0\) of a particle:\[K = E - E_0\]
  • Classically, kinetic energy is given by \(K = \frac{1}{2}mv^2\), but this changes with relativistic speeds.
  • As particles move closer to the speed of light, relativistic effects become significant.
This exercise demonstrates how kinetic energy is calculated following the understanding of both total and rest energies.
Rest Energy
Rest energy \(E_0\) is the inherent energy of an object even when it is at rest. It is derived from the relationship described by Einstein's famous equation \(E_0 = m_0 c^2\). This equation shows that mass can be seen as a form of energy.
  • Rest energy accounts for the entire energy an object has from its rest mass.
  • Even without movement, objects contain significant energy, reflected in massive particles.
In our calculations, the rest energy is necessary to find both kinetic energy and total energy, influencing our understanding of particle physics.
Relativistic Formulas
Relativistic formulas provide essential tools for analyzing particles moving at significant fractions of the speed of light. These equations connect various physics concepts such as energy, momentum, and mass.
  • Total energy is calculated using \(E = \sqrt{(p c)^2 + (m_0 c^2)^2}\), bundling momentum and rest energy together.
  • These formulas incorporate the speed of light \(c\) as it becomes essential near its upper speed limit.
  • They ensure accurate assessments of energies and movements for high-speed particles.
Understanding these formulas is key for assessing how various energy types are related and calculated, paving the way for learning advanced physics concepts.

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Most popular questions from this chapter

A spaceship is traveling toward earth from the space colony on Asteroid \(1040 \mathrm{~A}\) at a speed of \(0.9 \mathrm{c}\) relative to earth. Assume that the asteroid and earth are in the same frame of reference. When the ship is at the halfway point of the trip, as measured in the earth frame, two radio messages are sent simultaneously to the spaceship, one from earth and the other from 1040 A. Does the ship receive these two messages at the same time? Explain your reasoning.

An electron is acted upon by a force of \(5.00 \times 10^{-15} \mathrm{~N}\) due to an electric field. Find the acceleration this force produces in each case: (a) The electron's speed is \(1.00 \mathrm{~km} / \mathrm{s}\). (b) The electron's speed is \(2.50 \times 10^{8} \mathrm{~m} / \mathrm{s}\) and the force is parallel to the velocity.

Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes \(1.50 \mathrm{~s}\) as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment? (b) If each swing takes \(1.50 \mathrm{~s}\) as measured by a person at mission control on earth, how long will it take as measured by the astronaut in the spaceship?

Two events are observed in a frame of reference \(S\) to occur at the same space point, the second occurring \(1.80 \mathrm{~s}\) after the first. In a second frame \(S^{\prime}\) moving relative to \(S\), the second event is observed to occur \(2.35 \mathrm{~s}\) after the first. What is the difference between the positions of the two events as measured in \(S^{\prime} ?\)

At what speed is the momentum of a particle three times as great as the result obtained from the nonrelativistic expression \(m v ?\)
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