What is the ratio keafter/kebefore of the total kinetic energy after the collisions occur to that before the collisions occur?

is called square of Coefficient of Restitution

Explanation:

As we know that the coefficient of restitution is the ratio of the body after collision to the ratio of the body before collision.

Mathematically:

where:

relative velocity after collision

relative velocity before collision

also, in terms of Kinetic energy:

where:

relative kinetic energy after collision

relative kinetic energy before collision

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i hope this you out, have a wonderful day