Q.When two bodies make a head-on elastic collision, show that the coefficient of restitution equals 1.

The Bounce: What Makes a Ball Bounce Back?

Drop a rubber ball and it springs back up. Drop a lump of clay and it just splats. The difference between these two extremes is captured by a single number: the coefficient of restitution (often written as $e$).

The intuition is simple. When two objects collide, they always deform a little — like a spring getting squashed. The question is: how much of that "squash" energy gets given back as motion? A perfect spring gives everything back; a lump of clay gives nothing back. The coefficient of restitution measures exactly that.

The Precise Definition

For any collision along a straight line (what we call head-on or one-dimensional collision), the coefficient of restitution is defined as:

Let's unpack that. Suppose two balls, A and B, move toward each other. Before the collision, their velocities are $u_A$ and $u_B$. After the collision, they are $v_A$ and $v_B$.

The relative velocity of approach is how fast they are coming together: $u_A - u_B$ (taking direction into account — if they move toward each other, this is positive).

The relative velocity of separation is how fast they are moving apart after the collision: $v_B - v_A$.

So the formula becomes:

What the Numbers Mean

$e = 1$ — Perfectly Elastic Collision

No kinetic energy is lost. The relative speed of separation equals the relative speed of approach. Think of two billiard balls or ideal gas molecules. The "spring" gives back everything.

$e = 0$ — Perfectly Inelastic Collision

The objects stick together after impact. Their relative velocity of separation is zero. Think of two lumps of clay or a bullet embedding in a block. All the relative motion is lost.

$0 < e < 1$ — Partially Inelastic (Real World)

Every real collision falls here. A tennis ball against concrete might have $e \approx 0.7$; a basketball around $0.8$. Some kinetic energy is lost as heat, sound, and permanent deformation.

Why "Along the Line of Impact" Matters

The definition includes the phrase "along the line of impact." In a glancing collision (like a pool ball hitting another at an angle), the velocities have components both along and perpendicular to the line joining the centers. The coefficient of restitution applies only to the velocity components along that line. The perpendicular components obey a different rule (usually, they are unchanged if friction is negligible).

A Quick Example …