Foundations — Coefficient of restitution e = (v₂ − v₁) - (u₁ − u₂)

This is a foundations page. We will not solve collisions here — we will make sure that every letter and symbol used in the parent topic already means something to you before you meet it there.

1. A number line and a direction (the humblest symbol: the sign)

Before any velocity we need a way to say which way something moves. Physics does this with a number line: we pick one direction and call it "positive."

Look at the figure. The red arrow is our chosen positive direction. The size of a velocity tells you how fast; the sign tells you which way. Every velocity symbol in the whole topic secretly carries a sign — forget the sign and every collision answer will be wrong.

2. Speed vs velocity, and the letters $u$ and $v$

The topic uses two families of letters, and the choice is deliberate:

3. Relative velocity — the difference of two velocities

This is the single most important idea to master before the topic. When two things move on the same line, how fast is one closing in on the other?

In the figure both cars move right, but the back car (red) moves faster. Subtracting velocities is like sitting on the front car: from there the front car looks still, and the back car rushes toward you at exactly $u_1-u_2$. That is why a difference of velocities measures approach.

4. Approach and separation (and why the indices swap)

Notice the swapped order of the subscripts, and see why in the figure:

• For a crash to happen at all, the chaser (body 1) must be faster than the one ahead (body 2): $u_1>u_2$, so $u_1-u_2>0$. Writing it "$1$ minus $2$" keeps approach positive.
• After the crash, they must move apart, which means the front body (body 2) ends up faster: $v_2>v_1$, so $v_2-v_1>0$. Writing it "$2$ minus $1$" keeps separation positive.

Now we can assemble the symbol the whole topic is named after. The coefficient of restitution is simply separation divided by approach:

5. Mass, momentum, and why one equation isn't enough

This one equation contains two unknowns after a crash: $v_1$ and $v_2$. One equation, two unknowns — you cannot solve it. That gap is exactly the hole the coefficient of restitution fills: it is the second equation. See Conservation of Linear Momentum for the full story of the first one, and Impulse and Momentum for the deeper "impulse ratio" meaning of $e$.

6. Squares, roots, and the $\sqrt{h_2/h_1}$ shortcut

The bouncing-ball formula uses a square root ( ​). Here is why, from zero.

For a bouncing ball we track two heights, and it matters which is which:

Drop-height $h$ and impact speed $v$ are linked by $h=\frac{v^2}{2g}$, where $g\approx 9.8{\text{m/s}}^2$ is gravity's downward pull. Height depends on speed squared. So a ratio of heights is a ratio of speeds squared: $\frac{h_2}{h_1}={\left(\frac{v_{\text{rebound}}}{v_{\text{impact}}}\right)}^2=e^2\Rightarrow e=\sqrt{\frac{h_2}{h_1}}.$

How it all feeds the topic

The picture below is the "prerequisite map": follow the arrows and every foundation on this page flows into the single symbol $e$.

Read top to bottom: signs let us write velocities; velocities let us take differences (relative velocity); differences give approach and separation, whose ratio is $e$. Separately, mass gives momentum, whose conservation is the first equation that $e$ completes. Squares and roots convert the speed-ratio $e$ into a measurable height-ratio.

Once these pieces click, jump to the parent topic and the special cases Elastic Collisions ($e=1$) and Perfectly Inelastic Collisions ($e=0$); the energy side lives in Kinetic Energy Loss in Collisions.

Equipment checklist

Cover the right side and answer each aloud.