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A perfectly inelastic collision is when two objects stick together after colliding.

In a perfectly inelastic collision, the two objects involved stick together and move as one object after the collision. This means that the final velocity of the combined object is the same as the initial velocity of the two objects before the collision. The kinetic energy of the system is not conserved in a perfectly inelastic collision, as some of the energy is lost as heat and sound.

Mathematically, we can represent a perfectly inelastic collision using the conservation of momentum equation:

m1v1 + m2v2 = (m1 + m2)vf

where m1 and m2 are the masses of the two objects, v1 and v2 are their initial velocities, and vf is the final velocity of the combined object. Since the two objects stick together, their final velocity is the same, so we can simplify the equation to:

(m1 + m2)vf = m1v1 + m2v2

To solve for vf, we can divide both sides by (m1 + m2):

vf = (m1v1 + m2v2) / (m1 + m2)

This equation shows that the final velocity of the combined object is a weighted average of the initial velocities of the two objects, with the weights being the masses of the objects.

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