Does the total kinetic energy of an isolated system necessarily have to be conserved when its total momentum is conserved?
Momentum is conserved period.
total kinetic energy conserved? No, where did you get that idea?
Think about a bullet hitting a block and getting embedded. I am sure you have done that for homework. You solve it using conservation of momentum. However
(1/2)m V^2 of bullet
is not equal to
(1/2)(m + M) v^2 of bullet and block after
during the penetration, mechanical energy escaped as heat.
Thank for the explanation, I knew about momentum but the fact that kinetic energy was mentioned in the question just threw me off.
To determine whether the total kinetic energy of an isolated system is conserved when its total momentum is conserved, we must understand the conservation principles involved.
In an isolated system, both momentum and kinetic energy can be conserved separately or together, depending on the nature of the system and the interactions within it. Let's examine two scenarios:
1. Elastic Collision (Kinetic Energy Conservation + Momentum Conservation):
In an elastic collision, two objects collide and bounce off each other without any loss of kinetic energy. In this case, both the total momentum and total kinetic energy of the system are conserved. Elastic collisions typically involve objects with no deformation, such as billiard balls or gas molecules and are characterized by the conservation of both momentum and kinetic energy.
2. Inelastic Collision (Momentum Conservation, but not necessarily Kinetic Energy Conservation):
In an inelastic collision, two objects collide and stick together or deform upon impact, resulting in a loss of kinetic energy. In this case, although the total momentum of the system is conserved, the total kinetic energy is not necessarily conserved. Inelastic collisions can occur in real-world scenarios, such as car crashes, where some of the initial kinetic energy is transformed into other energy forms like heat or sound.
Therefore, while the total momentum is always conserved in an isolated system, the conservation of kinetic energy depends on the nature of the collision. In the case of an elastic collision, both kinetic energy and momentum are conserved, while in an inelastic collision, only the momentum is conserved, and kinetic energy may be lost.