Question

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

Short answer

No, the total energy of a system may or may not be conserved whenever its momentum is conserved.

Step-by-step solution

Definition of head on elastic collision

In a head-on elastic collision between a small projectile and a much larger target, the bullet will bounce back at nearly the same speed, while the big target will have a very low velocity. A ball bouncing back from the Earth when we throw it down is an example.

Explaining when total energy and momentum both are conserved  

When there is a head-on elastic collision between two spherical objects. The internal forces are generated in the form of action-reaction force between the objects during collision. According to the Newton’s third law action-reaction pairs are of same magnitude but opposite in directions.The action-reaction pairs acts on different bodies.

Therefore, if we consider a system of two bodies then the net external force on the system is zero. The force is defined as rate of change of momentum.So, when net force is zero , change in momentum will also be zero or momentum will remain conserve.

\(\begin{aligned}{F_{net}} &= \dfrac{{\Delta P}}{{\Delta t}} = 0\\\dfrac{{{P_{final}} - {P_{initial}}}}{{\Delta t}} &= 0\\{P_{final}} - {P_{initial}} &= 0\\{P_{final}} &= {P_{initial}}\end{aligned}\)

On the other hand total energy is the sum of kinetic energy plus potential energy. For elastic collision the value of coefficient of restitution is 1. So, there is no loss in the energy because there is no heat generation and no deformation in the shape of bodies. The objects restore their shapes after the collision.Thus, the total energy in case of elastic collision remains conserve.

So, in case of head-on elastic collision momentum as well as total energy of the system remain conserved.

 Condition in which momentum is conserved but total energy is not conserved 

An inelastic collision is one in which a part of kinetic energy is changed to some other form of energy like heat, deformation in shape.

As there is loss of kinetic energy in an inelastic collision because a part of initial kinetic energy get changed in to other form of energy.Therefore,the internal kinetic energy is not conserve in an inelastic collision.

While momentum of the system is conserved because all the forces generated between the bodies are internal forces and in the presence of internal forces momentum of the system remains conserve.

Thus, in case of an inelastic collision momentum of the system is conserved while the total energy of the system is not conserved due to loss in energy.

Therefore, there is no compulsion that the total energy must be conserved whenever momentum of the system is conserve.

In head-on elastic collision momentum and energy both are conserved while in case of an inelastic collision only momentum is conserved, not energy.

Hence, when momentum of the system is conserved, total energy may or may not be conserved.