Equivalence of Definitions of Perfectly Inelastic Collision

Theorem

The following definitions of the concept of Perfectly Inelastic Collision are equivalent:

A perfectly inelastic collision is a collision in which the coefficient of restitution is equal to zero.

A perfectly inelastic collision is a collision after which the bodies involved remain in contact with each other and move as one body.

Recall the definition of Coefficient of Restitution:

Let two solid bodies $B_1$ and $B_2$ be in collision.

Let the components of the relative velocities of $B_1$ and $B_2$ in the direction of their common normal be respectively:

$\mathbf u_1$ and $\mathbf u_2$ before the collision

$\mathbf v_1$ and $\mathbf v_2$ after the collision.

$\mathbf v_2 - \mathbf v_1 = -e \paren {\mathbf u_2 - \mathbf u_1}$

The coefficient $e$ is known as the coefficient of restitution.

Let $\CC$ be a perfectly inelastic collision by definition $1$.

Recall definition $1$ of Perfectly Inelastic Collision:

A perfectly inelastic collision is a collision in which the coefficient of restitution is equal to zero.

We have:

$\mathbf v_2 - \mathbf v_1 = -0 \times \paren {\mathbf u_2 - \mathbf u_1}$

That is:

$\mathbf v_2 - \mathbf v_1 = 0$

In other words:

$\mathbf v_2 = \mathbf v_1$

That is, $B_1$ and $B_2$ move with the same velocity, and as they have just collided, they are still in contact.

Thus $\CC$ is a perfectly inelastic collision by definition $2$.

$\Box$

Let $\CC$ be a perfectly inelastic collision by definition $2$.

Recall definition $2$ of Perfectly Inelastic Collision:

A perfectly inelastic collision is a collision after which the bodies involved remain in contact with each other and move as one body.

Then by definition:

$\mathbf v_2 = \mathbf v_1$

That is:

$\mathbf v_2 - \mathbf v_1 = 0$

We have:

$\mathbf v_2 - \mathbf v_1 = -e \times \paren {\mathbf u_2 - \mathbf u_1}$

But as $B_1$ and $B_2$ have just collided, it means that their initial velocities must have been unequal.

That is:

$\mathbf u_2 \ne \mathbf u_1$

That is:

$\mathbf u_2 - \mathbf u_1 \ne 0$

So for $-e \times \paren {\mathbf u_2 - \mathbf u_1} = 0$, it is necessary that $e = 0$.

Thus $\CC$ is a perfectly inelastic collision by definition $1$.

$\blacksquare$

2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): coefficient of restitution
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): coefficient of restitution