A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.

Certainly! When tutoring on UrbanPro, I always strive to provide clear explanations to help students understand complex concepts like this one. Let's break down the scenario.

In the laboratory frame of reference, if a nucleus at rest disintegrates into two smaller nuclei, we need to consider the principles of conservation of momentum and conservation of energy.

1. Conservation of Momentum: Before the disintegration, the total momentum of the system (the nucleus at rest) is zero since the nucleus is stationary. After disintegration, the total momentum must still be zero according to the law of conservation of momentum. If the products move in opposite directions, their momenta will cancel each other out, resulting in a net momentum of zero.

2. Conservation of Energy: The total energy before and after the disintegration must remain constant. Since the nucleus was at rest initially, its energy is purely rest energy (mc^2). After disintegration, this energy is distributed among the kinetic energies of the two smaller nuclei. However, the total kinetic energy must still equal the initial rest energy to comply with the conservation of energy.

Therefore, for the two smaller nuclei to move in opposite directions, they must have equal but opposite momenta and equal kinetic energies. This ensures that both momentum and energy are conserved in the process.

In tutoring sessions on UrbanPro, I often illustrate these concepts with examples and diagrams to make them more accessible to students, helping them grasp the underlying physics principles effectively.