Chapter 15: Problem 61
A particle of mass 3 MeV/c \(^{2}\) has momentum 4 MeV/c. What are its energy (in MeV) and speed (in units of \(c\) )?
Short Answer
Expert verified
The energy is 5 MeV and the speed is 0.8c.
Step by step solution
01
Define Relativistic Energy-Momentum Relationship
The relativistic energy-momentum relationship is given by \[E^2 = (pc)^2 + (m_0c^2)^2\]where:- \(E\) is the total energy- \(p\) is the momentum- \(m_0\) is the rest mass of the particle- \(c\) is the speed of light.
02
Substitute Known Values into the Formula
Given:- mass \(m_0 = 3 \, \text{MeV}/c^2\)- momentum \(p = 4 \, \text{MeV}/c\)Substituting into the formula:\[(E)^2 = (4 \, \text{MeV})^2 + (3 \, \text{MeV})^2\].
03
Calculate the Energy
Calculate \(E^2\):\[(E)^2 = 16 \, \text{MeV}^2 + 9 \, \text{MeV}^2 = 25 \, \text{MeV}^2\].Taking the square root gives:\[E = \sqrt{25 \, \text{MeV}^2} = 5 \, \text{MeV}\].
04
Use the Energy-Momentum Relationship for Speed
The speed \(v\) of the particle is given by the relationship \[v = \frac{pc^2}{E}\].
05
Substitute Values to Solve for Speed
Substitute the known values into the speed formula:\[v = \frac{(4 \, \text{MeV}/c) \cdot c^2}{5 \, \text{MeV}}\].Simplify to get:\[v = \frac{4}{5}c = 0.8c\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Energy-Momentum Relationship
The energy-momentum relationship is a fundamental principle in relativistic mechanics, allowing us to understand the link between a particle's energy, momentum, and rest mass. It is expressed mathematically as:\[E^2 = (pc)^2 + (m_0c^2)^2\]Here, \(E\) represents the total energy of the particle, \(p\) is its momentum, \(m_0\) is the rest mass, and \(c\) is the speed of light (approximately \(3 \times 10^8 m/s\)). This formula is crucial as it helps predict and calculate the total energy of particles moving at relativistic speeds—speeds close to the speed of light, where classical physics no longer applies.
- \((pc)^2\) is the energy due to momentum alone.
- \((m_0c^2)^2\) is the contribution due to the rest mass energy.
Relativistic Energy
Relativistic energy takes into account the total energy possessed by an object moving at speeds that are substantial fractions of the speed of light. Conventional Newtonian physics, which considers energy solely based on mass and velocity, fails at such high velocities. Here, relativistic effects become significant.The rest energy, given by \(E_0 = m_0c^2\), represents the inherent energy an object possesses due solely to its mass. When a particle moves at relativistic speeds, its total energy is a combination of this rest energy and additional energy from its momentum. This additional energy is a non-negligible part of the total energy, contrary to classical scenarios.In our example, where a particle has a rest mass of 3 MeV/c\(^2\) and momentum of 4 MeV/c, substitution yields \[(E)^2 = (4 \, \text{MeV})^2 + (3 \, \text{MeV})^2 = 25 \, \text{MeV}^2\]Solving this provides a total energy of 5 MeV, demonstrating the combination of kinetic and rest mass energies.
Particle Speed Calculation
Determining the speed of a particle in relativistic mechanics involves not just simple velocity calculations, but understanding its energy and momentum relationships. The speed \(v\) can be computed using the equation:\[v = \frac{pc^2}{E}\]In the example exercise, with a momentum of 4 MeV/c and energy of 5 MeV, substituting these values gives:\[v = \frac{(4 \, \text{MeV}/c) \cdot c^2}{5 \, \text{MeV}}\]This simplifies to:\[v = \frac{4}{5}c = 0.8c\]Because the speed of light \(c\) is the ultimate speed limit, relativistic speed computations often yield results as fractions of \(c\). Here, the calculated speed of the particle is 0.8c, highlighting significant relativistic effects, as the speed is a substantial fraction of the speed of light. Such considerations are essential for accurately determining how particles behave at high velocities, affording insight into the dynamics of fast-moving particles often seen in high-energy physics and particle accelerators.