Stun Shot — Cue Ball Speed

The collision cannot touch the component of the cue ball's velocity that lies ACROSS the line of centres, so the cue ball keeps it: $v_{CB} = u\sin\varphi$, directed along the tangent line — the line through the contact point perpendicular to the line of centres. Put that beside the object ball's $u\cos\varphi$ along the line of centres and you have the most famous result in the game.

The 90° rule, and here is the proof rather than the slogan. One velocity is $u\cos\varphi$ along the line of centres. The other is $u\sin\varphi$ along the tangent line. Those two directions are perpendicular by construction, so the paths are at 90° for every cut angle there is. Square them and add: $u^2\cos^2\varphi + u^2\sin^2\varphi = u^2$. Kinetic energy balances exactly, momentum balances by components, and nothing has been assumed except equal masses and a smooth elastic contact. The 90° rule is not an approximation and not a coaching heuristic. It is Pythagoras.

And it holds only for a stun shot. This is the honesty that has to sit right next to the rule, because the rule is taught far more often than its condition is.

"Stun" means the cue ball arrives at the object ball SLIDING, with no top or bottom spin left on it — its centre is moving but its surface is not rolling. Only then is the cue ball's post-collision motion just $u\sin\varphi$ along the tangent line and nothing else. Give the cue ball follow, and after the collision it is still carrying forward spin; friction converts that spin back into travel, and the cue ball curves FORWARD off the tangent line. Give it draw, and it curves BACK. In both cases the angle between the two balls' final paths is not 90°, and can be nowhere near it.

Applying the 90° rule to a follow shot is the classic mistake of the player who has just learned some physics, and it makes them worse for a while than the player who learned nothing. The rule is exactly right in a narrow case and confidently wrong outside it. The cure is to know how long the stun lasts: a cue ball struck with no spin slides for a distance $12v_0^2/49\mu g$ and then it is rolling, and past that point you want the rolling-deflection page instead, not this one.

The other classic mistake is getting the two lines backwards. Say it out loud until it sticks: the tangent line is where the CUE ball goes; the line of centres is where the OBJECT ball goes. They are at right angles, so mixing them up does not put you slightly off — it sends you to the wrong side of the table entirely. Everyone does it once. The mnemonic that works is that the object ball is pushed, and you can only push something in the direction of the line between the two centres.

What the rule is FOR is position, and that is where its value lies. On a stun cut you know, without any calculation at all, that your cue ball is leaving perpendicular to the object ball's path — so if the object ball is heading for a corner pocket, look 90° off that line and you know your cue ball's road. Many players sight it with two fingers held in a V, one down each path. It is the single cheapest piece of shot-planning in the game, and it costs nothing but remembering that it needs a stun.

A footnote on where this comes from. Coriolis worked all of this out and published it in 1835, in Théorie mathématique des effets du jeu de billard — the same Coriolis of the rotating-frame force, applying his mechanics to a game he clearly loved. It sat largely unread by players for a century and a half. Almost everything modern instruction says about the tangent line, the 30° rule and natural roll is in that book, in French, forty years before the first world championship.