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In elastic collisions, such as the collisions of two snooker balls, it is stated that conservative forces are responsible for the collision being elastic.

(See http://vlab.amrita.edu/?sub=1&brch=68&sim=189&cnt=1)

Why is it that these elastic collisions have to be due to conservative forces?

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I think you would be better of asking the question over here. That being said:

Since the energy is conserved, by definition, the total work done is $0$ (since $W=\Delta E_k=0$). Now, the work $W$ along a path $S$ is given by $$W=\int_S\mathbf{F}\cdot d\mathbf{r}=0$$ so the forces are conservative.

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