Must the total energy of a system be conserved whenever its momentum is conserved? Explain why or why not.

It depends on the type of momentum that is being considered when making the calculations. Generally when the external force acting on the system is zero - there are no external influences - there is a conservation of the momentum of the system. This is appreciated with greater amount, when considering an elastic system, for which the momentum of the system and the kinetic energy are conserved.

However, in inelastic collisions, the momentum is usually conserved but the kinetic energy is not. This happens because the system can transfer the kinetic energy in other types of energy to its surroundings such as thermal energy for example. This causes that energy in the system to decrease.

Therefore energy is not always conserved when the momentum is conserved.