Question

A positron and an electron annihilate, producing two 2.0 -MeV gamma rays moving in opposite directions. Calculate the kinetic energy of the electron when the kinetic energy of the positron is twice that of the electron.

Short answer

Answer: The kinetic energy of the electron is approximately 0.993 MeV.

Step-by-step solution

Understand the Given Information

We are given that a positron and an electron annihilate, and two gamma rays, each with an energy of 2.0 MeV, are produced. The kinetic energy of the positron is twice that of the electron. We need to find the kinetic energy of the electron.

Apply Conservation of Momentum

Since there is no external force acting on the system, the total momentum before the annihilation is equal to the total momentum after the annihilation. Let electron initial momentum be p_e and positron initial momentum be p_p. The initial momentum of gamma rays is 0 since they are produced after the annihilation. Total initial momentum = p_e + p_p Total final momentum = -p_gamma1 + p_gamma2 = 0 (since gamma rays move in opposite directions) So, p_e + p_p = 0 or p_p = -p_e

Apply Conservation of Energy

The total energy of the system must be conserved before and after the annihilation. Let KE_e be the kinetic energy of the electron and KE_p be the kinetic energy of the positron. Initial total energy = KE_e + KE_p + rest energy of electron and positron = KE_e + KE_p + 2*(0.511 MeV) Final total energy = energy of two gamma rays = 2 * 2 MeV = 4 MeV Conservation of energy equation becomes: KE_e + KE_p + 1.022 MeV = 4 MeV

Use the Given Relation Between Kinetic Energies

We know that the kinetic energy of the positron is twice that of the electron (KE_p = 2*KE_e). Now we can rewrite the conservation of energy equation by substituting the given relation: KE_e + 2*KE_e + 1.022 MeV = 4 MeV

Solve for the Kinetic Energy of the Electron

Combine the terms in the equation and solve for KE_e: 3*KE_e + 1.022 MeV = 4 MeV 3*KE_e = 4 MeV - 1.022 MeV 3*KE_e = 2.978 MeV KE_e = 2.978 MeV / 3 KE_e ≈ 0.993 MeV Hence, the kinetic energy of the electron is approximately 0.993 MeV when the kinetic energy of the positron is twice that of the electron.

Key concepts

Conservation of Momentum

In physical processes where particles interact with each other, one fundamental principle that always holds true is conservation of momentum. This principle states that the total momentum of a closed system remains constant if it is not affected by external forces. In the context of electron-positron annihilation, before the particles collide, they each have momentum, equal in magnitude but opposite in direction. After they collide, the resulting gamma rays carry away the system's momentum.

Momentum is a vector quantity, which means it has both magnitude and direction. For the electron ($p_e$) and the positron ($p_p$), annihilating to produce gamma rays, the sum of their momenta equals zero ($p_e+p_p=0$). It’s essential for students to grasp this concept because it helps to predict the behavior of particles after a collision without knowing the detailed properties of the particles or the force that caused the interaction.

Conservation of Energy

The law of conservation of energy is another cornerstone of physics, especially in particle interactions. This law indicates that the total energy in an isolated system remains constant over time. During the electron-positron annihilation process, the energy includes the kinetic energy of both particles and their rest mass energy, which is the energy equivalent to their mass according to Einstein's famous equation, E=mc^2.

In the case at hand where an electron and a positron annihilate, the energy transformed into the gamma rays originated from both the kinetic energies of the particles ($K{E}_e+K{E}_p$) and their rest energies. Combining these gives us the total initial energy, which will be equal to the final energy carried by the gamma rays. This example presents a direct application of the conservation of energy principle and underscores its importance in calculating the outcomes of particle interactions.

Kinetic Energy

Kinetic energy is the energy a particle possesses due to its motion, and it is directly proportional to the mass and square of the speed of the particle. Understanding kinetic energy is pivotal in physics problems, especially those involving motion or collisions. In our electron-positron annihilation problem, the kinetic energies of both the electron ($K{E}_e$) and the positron ($K{E}_p$), along with their rest mass energies, determine the total energy available in the system before annihilation.

The textbook problem specifies that the positron's kinetic energy is twice that of the electron ($K{E}_p=2\times K{E}_e$). By employing this relationship, we can simplify the conservation of energy equation to solve for the unknown kinetic energy of the electron. Recognizing the role of kinetic energy in such interactions allows students to analyze and predict the dynamics of matter under various conditions.

Gamma Rays

Gamma rays are high-energy electromagnetic radiation and are a product of certain types of nuclear and particle reactions, including electron-positron annihilation. They are significant because they indicate the amount of energy released during such events, and they are employed in various applications, from medical treatments to the study of the universe.

In our annihilation scenario, each gamma ray carries an energy of 2.0 MeV. The fact that there are two gamma rays moving in opposite directions is what makes this example a perfect demonstration of both conservation of momentum and energy. It is important for students to understand the characteristics of gamma rays since they often signify the conversion of particle mass into pure energy, a key concept in both astrophysics and particle physics.