# When is mechanical energy ever conserved for inelastic collisions?

So as I was taking a test today I encountered an intriguing question, the question came in two parts. The first parts, which I answered quite easily, was about whether or not mechanical energy is conserved in an inelastic collision if the system, consisting of two objects, is closed and isolated on earth. I knew that it was not conserved since some heat is produced following the perfectly inelastic collision of the two objects. However, the second part of the question is what left me uncertain. The second part of the question asked whether or not mechanical energy is conserved if the system is earth and the two objects colliding perfectly inelastically afterwards. I was uncertain of the definition of the term mechanical energy.

• link see if this helps. :) Apr 25, 2018 at 23:25
• If I have 2 blocks and a 'perfect-massless' spring: they can collide, compress the spring, lock it, and the collision is inelastic. Yet my imaginary spring has dissipated no energy-it's stored (mechanically) in the spring.
– JEB
Apr 26, 2018 at 1:01
• @JEB The act of locking requires either external energy or the taking of energy from the objects, so it is not elastic. Apr 26, 2018 at 2:17
• I don't understand your penultimate sentence. Can you clarify what you mean by "the system is Earth and the two objects". Are the two objects hitting the ground, or ... ??. Mechanical energy is kinetic energy and potential energy. It excludes thermal energy, radiant energy, acoustic energy, etc. Apr 26, 2018 at 2:21
• @garyp the scenario is like this, two blocks are sliding on a frictionless ground on earth, they collide and stick together. I am really asking that does the settings of the system change the fact of whether or not mechanical energy is conserved or not. Apr 26, 2018 at 3:17

Like in this https://www.khanacademy.org/science/physics/linear-momentum/elastic-and-inelastic-collisions/a/what-are-elastic-and-inelastic-collisions, in an inelasatic collision,the particles do not regain their shape and size completely after collision .Some fraction of the mechanical energy is retained by the colliding particles in the form of deformation potential energy.Thus,Kinetic energy of the particles no longer remains conserved anf gets converted into something else which is usually heat.However in absence of external forces,law of conservation of linear momentum still holds good.

Collisions are said to be perfectly elastic if the particles stick together and move with same velocity.Momentum is of course conserved here but maximum kinetic energy is lost in this case.

Like for example take a bullet getting stuck on a block hanging on a spring.When the bullet hits the block then the bullet and block would become hot. This is still a form of kinetic energy in reality, but it's now the random jiggling of molecules. This is not considered mechanical energy since it is not easily used for mechanical purposes.

• Thanks for the effort! I understand that the objects do not retain their original shapes after an inelastic collision, however, this is not what I am asking. I am asking whether or not mechanical energy is conserved if the setting of the system changes. Apr 26, 2018 at 3:20

The loss of mechanical energy is not dependent on the setting.

Total energy is conserved in collisions. In elastic collisions the combined KE of the colliding objects remains unchanged by the collision. In an inelastic collision some of the kinetic energy of the colliding objects is converted into other forms of energy, such as sound and heat, so the KE is reduced.

The sound and heat energy is not classed as mechanical energy, because it is effectively dissipated and can no longer be used to do work.

Typically, mechanical energy refers to kinetic energy of interacting bodies and does not include the internal energy of the bodies. This comes from the physics mechanics definition of work as a force acting through a distance to produce a change in kinetic energy. Point particles have no internal energy and rigid bodies have constant internal energy (no "heating" effects). For an inelastic collision between non-rigid bodies the internal energy of the bodies can change and mechanical (total kinetic) energy is not conserved. Bottom line: for inelastic collisions mechanical energy is not conserved because of changes in internal energy.

Thermodynamics defines work as "energy that crosses the boundary between a system and its surroundings without transfer of mass due to an intensive property difference other than temperature". Heat is "energy that crosses the boundary between a system and its surroundings without transfer of mass solely due to a difference in temperature". This broader definition of work includes forms other than mechanical work, such as electrical current crossing a boundary.

In general, to evaluate interactions where internal energy is not constant the first law is needed. For example, the first law is needed to evaluate the change in internal energy of a non-rigid body slipping under friction that "heats up" (increases internal energy) and transfers heat to the surroundings.

I was uncertain of the definition of the term mechanical energy.

The mechanical energy is defined as the sum of both the kinetic and potential energy of a system.

The second part of the question asked whether or not mechanical energy is conserved if the system is earth and the two objects colliding perfectly inelastically afterwards.

As you correctly pointed out, after the inelastic collision the kinetic energy transformed into heat energy. If you regard your system as being the Earth and the two colliding objects (which are stuck), there is no energy crossing the system boundary. This means that the energy of the system is conserved.

However, the mechanical energy is not conserved due to the fact that the kinetic energy is not the same before and after the collision. This fact implies that mechanical energy is not the same before and after the collision (I am working with 1D collisions so that potential energy is zero before and after the collision). Note that KE transformed into heat energy, which is not included into the mechanical energy definition.