# How the numbers were worked out

Every exit line and number in this guide comes from one small, standard model of a ball bouncing with friction. Here is what it assumes and what it leaves out.

## The model

- The ball is treated as a rigid sphere at the moment of contact. Its squash shows up only through the bounce factor.
- Straight out of the wall, the ball keeps a set share of its speed: the bounce factor.
- Along the wall, friction pushes against the sliding of the ball's surface, up to a set share of the push-out (the grip). If that is enough to stop the sliding, the ball leaves rolling; if not, it slides all the way through.
- How easily the ball turns is set by how its weight is spread: a hollow ball (most of its weight in its skin) turns less easily than a solid one of the same weight.
- The flight between walls has gravity but no air: no drag and no bend from spin. Where the guide talks about the bend in the air, it says so in words and draws it by hand.

## The settings

- The middle ball, used in most figures: bounce factor 0.7, grip 0.5 on walls and bounce factor 0.75, grip 0.5 on the floor.
- The lively ball: bounce factor 0.85, grip 0.5. The dead ball (or a very hard hit): bounce factor 0.55, grip 0.5.
- The hand: bounce factor 0.5, grip 0.9 (a hand is soft and grippy).
- A "strong" spin is one whose surface moves at 30% of the ball's speed.

These settings are reasonable guesses, not measurements of handballs on handball walls. They are chosen to show the direction and rough size of each effect. No setting tried (bounce factors from 0.55 to 1, grips from 0.2 to 0.8, hollow and solid balls) reversed any direction this guide gives, though with little grip some effects shrink almost to nothing. The sizes do change with the settings, so read every number as "about" and "for this case".

## What it leaves out

- The air: a real ball slows down and bends in flight, so real shots land shorter than the figures, and spin bends them as well.
- The bounce factor falling as the ball hits harder (the guide describes it in words).
- A very grippy ball springing back past rolling, which can reverse a little of its spin. The model can do this but the guide's numbers leave it out.
- Uneven walls, dust, damp, cold balls and worn balls.

## The script

The model is a short program in this site's source: web/scripts/guide/bounceModel.ts, with tests (bounceModel.test.ts) that check each direction this guide states. Run web/scripts/guide/bounceTables.ts to print its tables of exit angles.

This kind of model is the standard textbook treatment of a ball bouncing with friction. Real balls are more complicated, and a coach's eye or a high-speed camera is the final word.

Part of the angles guide: https://handball.theperfectship.com/guide/angles.md