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(D) In a one-dimensional elastic collision,the coefficient of restitution $e$ is defined as the ratio of the magnitude of relative velocity of separat

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Both $(B)$ and $(C)$

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(D) In a one-dimensional elastic collision,the coefficient of restitution $e$ is defined as the ratio of the magnitude of relative velocity of separation to the magnitude of relative velocity of approach.$e = \frac{|\vec{v}_{sep}|}{|\vec{v}_{app}|} = \frac{|\vec{v}_2|}{|\vec{v}_1|}$.For an elastic collision,$e = 1$,which implies $|\vec{v}_2| = |\vec{v}_1|$. Since the direction of relative velocity reverses after an elastic collision,we have $\vec{v}_2 = -\vec{v}_1$,or $\vec{v}_1 = -\vec{v}_2$.For an inelastic collision,$0 \leq e < 1$,which implies $|\vec{v}_2| = e|\vec{v}_1|$. Since the direction of relative velocity also reverses in inelastic collisions,$\vec{v}_2 = -e\vec{v}_1$,which can be written as $\vec{v}_1 = -\frac{1}{e}\vec{v}_2$. Let $k = \frac{1}{e}$. Since $0 \leq e < 1$,$k \geq 1$. Thus,$\vec{v}_1 = -k\vec{v}_2$ holds for all collisions where the relative velocity reverses direction.

In a one-dimensional collision between two particles,their relative velocity is $\vec{v}_1$ before the collision and $\vec{v}_2$ after the collision.

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