How Do Elastic, Inelastic, and Explosion Collisions Differ?

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An elastic collision is characterized by the conservation of kinetic energy, meaning that the total energy of the colliding objects remains unchanged before and after the collision. In contrast, inelastic collisions result in energy loss due to factors such as friction and deformation, leading to a decrease in the total kinetic energy. A perfectly inelastic collision represents the extreme case where the maximum energy is lost, and the colliding objects stick together post-collision. An explosion collision can be viewed as a perfectly inelastic collision occurring in reverse, where energy is gained. Understanding these distinctions is crucial for analyzing different types of collisions in physics.
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What is an elastic collion? I don't understand the difference between elastic, inelastic and explosion collisions.
 
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In an elastic collision, no energy is lost. This means that the collective energy of the objects that collide, before the collision and after, is the same. In other collisions, energy is lost due to things like friction, deformation of the objects, sound, etc.
 
An explosion collision is a perfectly inelastic collision run in reverse. A perfectly inelastic collision represents the maximum amount of energy that can be lost (or gained, in this case) during the collision.

Elastic: No energy is lost by the colliding objects. Maximum boingy.

Inelastic: Energy is lost by the colliding objects.

Perfectly inelastic: The maximum amount of energy is lost by the colliding objects. Sticky.
 
I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
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