Energy Lost During Purely Inelastic Collisions

[jon4444]

I'm curious about how the math comes out when you apply conservation of momentum to the theoretical problem of a moving object having a purely inelastic collision with another stationary object in a single dimension. Since the velocity of the combined object is entirely determined by the initial speed of the moving object and the masses of the objects, these parameters also determine the amount of energy lost (when you compare the kinetic energy of the initial object versus that of the combined object).
But shouldn't energy loss be related to how much smashing and deformation goes on during the collision? Is there another interpretation I'm missing?

 

[tnich]

I'm curious about how the math comes out when you apply conservation of momentum to the theoretical problem of a moving object having a purely inelastic collision with another stationary object in a single dimension. Since the velocity of the combined object is entirely determined by the initial speed of the moving object and the masses of the objects, these parameters also determine the amount of energy lost (when you compare the kinetic energy of the initial object versus that of the combined object).
But shouldn't energy loss be related to how much smashing and deformation goes on during the collision? Is there another interpretation I'm missing?

You have assumed that the two objects smash together and then continue to move together as one object. The energy lost to the smashing and deformation required to reach that state is exactly equal to the kinetic energy loss.

 

[Dale]

these parameters also determine the amount of energy lost (when you compare the kinetic energy of the initial object versus that of the combined object).

Yes.

shouldn't energy loss be related to how much smashing and deformation goes on during the collision?

Yes. The two amounts are the same.

 

[Merlin3189]

But shouldn't energy loss be related to how much smashing and deformation goes on during the collision?

But shouldn't smashing and deformation be related to energy loss? Or even, determined by it.

 

[jon4444]

so, say certain materials require a certain amount of energy to join together (the smashing and deformation)--this essential sets a critical condition for if you could have an inelastic condition (i.e., only under certain relative masses and initial speed).
Is that a correct interpretation of the situation?

 

[Dale]

so, say certain materials require a certain amount of energy to join together (the smashing and deformation)--this essential sets a critical condition for if you could have an inelastic condition (i.e., only under certain relative masses and initial speed).
Is that a correct interpretation of the situation?

Yes, for example, if a material has an elastic region and a plastic region in its stress strain curve then you would not get a plastic collision at low energies.

 

[neilparker62]

Since the OP asked about the maths, then (for the record) energy loss in a perfectly inelastic collision is given by:

$ΔE = ½μΔv^2$

where μ is the reduced mass of the colliding objects and Δv their relative velocity.