Why Is the Coefficient of Restitution Calculated as (Va - Vb)/(Vb - Va)?

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SUMMARY

The coefficient of restitution is calculated as (Va - Vb)/(Vb - Va) to ensure a positive value representing the ratio of relative velocities during collisions. This formula accounts for the initial velocities (Va and Vb) of two colliding objects and their final velocities (Ua and Ub), emphasizing that these velocities are not necessarily equal. The choice of notation and order in the calculation is crucial for obtaining meaningful results, particularly in distinguishing between elastic and inelastic collisions.

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Why is it that the co-efficient of restitution is given by

(Va - Vb)/(Vb - Va) and not (Va - Vb)/(Va - Vb)?
 
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Depends much on how you decide to orient the reference system... anyway pay attention that initial velocities and final velocities are not necessarily equal, therefore you have better using different notations (like Va and Vb for initial velocities of the two objects and Ua and Ub for final ones)... indeed in your notation the second case you wrote would give you one no matter what happens...
 
But why do I use the Va- vb and then in the denominator use Ub-Ua?
 
http://en.wikipedia.org/wiki/Coefficient_of_restitution

As far as I understand, you want to have a number between 0 and 1 (0 being the case in which one object stops, 1 being elastic collision)... then you have to do the ratio of the final relative velocities and initial relative velocities... this order is simply used to get two positive numbers... you could also put absolute values and get the same result with any order...
 

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