Question

Particle accelerators fire protons at target nuclei so that investigators can study the nuclear reactions that occur. In one experiment, the proton needs to have 20 MeV of kinetic energy as it impacts a 207Pb nucleus. With what initial kinetic energy (in MeV) must the proton be fired toward the lead target? Assume the nucleus stays at rest. Hint: The proton is not a point particle.

Accepted Answer

Understand the problem: The proton must have 20 MeV of kinetic energy when it impacts the 207Pb nucleus. However, the proton is not a point particle, so we must account for the fact that the proton's charge interacts with the electric field of the lead nucleus. This interaction creates a repulsive electrostatic potential energy that the proton must overcome to reach the nucleus. Step 1: Calculate the electrostatic potential energy between the proton and the lead nucleus at the point of impact. Use the formula for electrostatic potential energy: U = k_e ($$q_1$$ $$q_2) / r$$, where k_e is Coulomb's constant, $$q_1$$ and $$q_2$$ are the charges of the proton and lead nucleus, and r is the distance of closest approach (approximately the sum of the radii of the proton and lead nucleus). Step 2: Determine the total energy required for the proton to reach the lead nucleus. The total energy is the sum of the proton's desired kinetic energy at impact (20 MeV) and the electrostatic potential energy calculated in Step 1. This total energy represents the initial kinetic energy the proton must have when fired. Step 3: Express the initial kinetic energy of the proton as K_{initial} = K_{final} + U, where K_{final} is the desired kinetic energy at impact (20 MeV) and U is the electrostatic potential energy. Substitute the values from Step 1 and Step 2 into this equation. Step 4: Solve for the initial kinetic energy K_{initial}. This value represents the energy the proton must have when fired to overcome the electrostatic repulsion and still retain 20 MeV of kinetic energy upon impact with the lead nucleus.

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Concetti chiave

Kinetic Energy

Conservation of Momentum

Rest Mass and Relativistic Effects