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How does the center of mass of a collision change after a collision has occurred compared to before. For instance, I know that it is moving, but does the velocity change and if so does it depend on whether the collision is elastic or inelastic

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If you have a system which consists of two blocks then the velocity in a given direction of the centre of mass of the system of two blocks does not charge if there are no external forces acting on the blocks in that direction.

So if the blocks are moving on a flat and friction-less horizontal table the horizontal velocity of the centre of mass of the two blocks will not change irrespective of what type of collision they undergo.

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The collision is said to happen in an instant, changing the velocity and rotational speed of a body, but the position and orientation do not change.

The change in velocity can be expressed as $$ \Delta v = \frac{1}{m} \int F\,{\rm d} t $$

where the force $F$ is short lived and very high in value such as the impulse $J = \int F \, {\rm d}t$ is a finite value. The result is that $\Delta v$ is finite.

But the change in position is zero because the finite velocity is multiplied by an infinitesimally small time

$$ \Delta x = \int v\, {\rm d}t \rightarrow 0 $$

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