Why is the loss of energy maxed in perfectly inelastic collision? I was doing a physics problem, and it had asked me to find the maximum $K_e$ loss, but I don’t understand why the loss is maxed when the objects stick together (the collision is inelastic).
 A: If one moves into the center of mass frame of the incoming particles/objects, for a perfectly inelastic collision the relative velocity gets completely quenched. The kinetic energy corresponding to this difference in velocity gets converted to heat, elastic energy of the deformed objects, sound etc. Due to momentum conservation, the net momentum of the center of mass (not visible in the center of mass frame anyway) can not change in the absence of external forces. Consequently, the total loss of the relative velocity is the best that can be done. On the other hand, in a perfectly elastic collision, the magnitude of the relative velocity remains unchanged, and no energy is lost.
This change in the relative velocity is parametrized by the coefficient of restitution $(e)$: https://en.wikipedia.org/wiki/Coefficient_of_restitution.
You can try and show that the energy loss has the expression $$\Delta E=\frac{1}{2}\frac{m_1m_2}{m_1+m_2}(1-e^2)\Delta v^2$$ where $m_1,m_2$ are the masses of the colliding objects and $\Delta v$ is the initial relative velocity.
So, one can smoothly span the entire range of parameter to tune how much energy is lost.
A: First of all you should know what are the events occurring during collision.
Let us imagine two spherical bodies of mass $m_1$ and $m_2$ moving with velocities$v_1$ and $v_2$ respectively ($v_1 > v_2$). Now , we know the two bodies are going to collide.

Now , one thing that is important to note here is that during collision bodies in contact get deformed since there is nothing like a perfect rigid body. Now what happens after deformation depends upon the material of the two bodies in collision. They can be either classified as an elastic or a  plastic or inelastic body. You can visualise elastic properties by taking the example of a spring.

1. Collision between elastic bodies
When elastic bodies collide some of the kinetic energy of each of the bodies is utilised in deforming the bodies and this deformation energy gets stored in the form of elastic potential energy. After maximum compression, the two bodies start separating and being elastic, they regain their shape and the stored potential energy is transformed into kinetic energy and almost negligible amount of energy is lost.

2. When the two bodies are perfectly plastic or inelastic
As you know they will deform. And the kinetic energies are utilised in deforming each other. But this time , there is no elastic potential energy and only deformation energy. Being plastic they can't apply reshaping force and thus the bodies stick together and move with a common velocity after maximum deformation. Since you can't get your kinetic energy back , it is better to say that the bodies lost their kinetic energies.

3. When the two bodies are semi elastic
If the two bodies are semi elastic i.e possess both the characters of a plastic and an elastic body. So this time , some of the kinetic energies are stored as elastic potential energy and some of them are utilised in permanent deformation.  And after collision they have a little bit of deformation.

So from the above three cases , it is pretty clear that the loss of kinetic energy is maximum when the deformation is maximum and this occurs when collison is perfectly plastic or inelastic.
Hope it helps .
