An electron and a positron are moving toward each other and each has speed 0.500c0.500c in the lab frame. What is the kinetic energy of each particle?

Ch 38: Photons: Light Waves Behaving as Particles

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 38: Photons: Light Waves Behaving as Particles

Problem 22a

Chapter 38, Problem 22a

An electron and a positron are moving toward each other and each has speed $0.500 c$ in the lab frame. What is the kinetic energy of each particle?

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Step 1: Recall the formula for relativistic kinetic energy, which is given by $ KE = (\gamma - 1)m c^2 $, where $ \gamma $ is the Lorentz factor, $ m $ is the rest mass of the particle, and $ c $ is the speed of light.

Step 2: Calculate the Lorentz factor $ \gamma $ using the formula $ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $, where $ v = 0.500c $. Substitute $ v $ into the equation to find $ \gamma $.

Step 3: Use the rest mass of the electron (or positron), which is $ m = 9.11 \times 10^{-31} \; \text{kg} $, and the speed of light $ c = 3.00 \times 10^8 \; \text{m/s} $. Substitute these values into the kinetic energy formula along with the calculated $ \gamma $.

Step 4: Simplify the expression $ KE = (\gamma - 1)m c^2 $ by substituting the numerical values for $ \gamma $, $ m $, and $ c $. This will give the kinetic energy of one particle.

Step 5: Since the problem asks for the kinetic energy of each particle, note that the calculation is the same for both the electron and the positron because they have the same mass and speed. The result from Step 4 applies to both particles.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relativistic Kinetic Energy

In relativistic physics, the kinetic energy of an object moving at a significant fraction of the speed of light (c) is given by the formula KE = (γ - 1)mc², where γ (gamma) is the Lorentz factor. The Lorentz factor accounts for time dilation and length contraction effects at high speeds, defined as γ = 1 / √(1 - v²/c²). This concept is crucial for accurately calculating the kinetic energy of particles like electrons and positrons moving at relativistic speeds.

Lorentz Factor

The Lorentz factor (γ) is a key component in the theory of special relativity, representing how much time, length, and relativistic mass increase as an object approaches the speed of light. It is calculated using the formula γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. Understanding the Lorentz factor is essential for determining the relativistic effects on the kinetic energy of particles moving at high speeds.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant over time. In the context of particle physics, this means that the kinetic energy of particles, such as electrons and positrons, must be accounted for when analyzing their interactions. This principle is fundamental when calculating the energies involved in particle collisions and transformations, ensuring that all forms of energy, including kinetic and rest mass energy, are considered.

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