Rolling Cue Ball Deflection (the real 30° rule)

This is the page the "30° rule" deserves, because the rule is a rule of thumb and the real thing is a curve — and the curve is more interesting than the rule.

Set the shot up. The cue ball arrives in full natural roll, so its centre is moving at $u$ and its surface is rolling to match. It strikes the object ball at cut angle $\varphi$. The collision is fast — a couple of hundred microseconds — and it takes away the component of velocity along the line of centres, leaving the cue ball moving at $u\sin\varphi$ along the tangent line. But the collision does essentially nothing to the cue ball's SPIN, which is still turning as though the ball were rolling forward along its original direction at $u$.

So immediately after contact the cue ball is a sliding ball again — moving one way, spinning as if to go another. Cloth friction then does what it always does, and the natural-roll result tells us the answer without integrating anything, because that result works just as well on vectors as on scalars:

\[\vec{v}_f = \tfrac{5}{7}\vec{v}_{\text{after}} + \tfrac{2}{7}\vec{v}_{\text{spin}}\]

The first term is $u\sin\varphi$ along the tangent line. The second is $u$ along the ORIGINAL direction of travel, because that is the motion the surviving spin corresponds to. Resolve both onto the tangent line and its normal, and the answer is remarkably tidy: the tangential part becomes $\tfrac{5}{7}u\sin\varphi + \tfrac{2}{7}u\sin\varphi = u\sin\varphi$, and the normal part is $\tfrac{2}{7}u\cos\varphi$. Divide:

\[\tan(\text{angle off the tangent line}) = \tfrac{2}{7}\cot\varphi\]

That is the classical result, and it is where the 2/7 you met on the natural-roll page reappears. Measured instead from the cue ball's ORIGINAL line of travel — which is what a player cares about, and what this page's $\alpha$ is — the same geometry gives $\tan\alpha = 5\sin\varphi\cos\varphi/(2 + 5\sin^2\varphi)$, the two being related by $\alpha = 90° - \varphi - \arctan(\tfrac{2}{7}\cot\varphi)$.

Now run the numbers, because this is the whole point of the page.

Between 20° and 40° of cut — which is where an enormous fraction of real shots live — the deflection wanders between 31.2° and 33.7° and never leaves that band. Two and a half degrees of variation across twenty degrees of cut angle. That plateau is why the 30° rule works, and it is a much better rule than it has any right to be. It is not that the deflection happens to be 30°; it is that the function has a broad maximum right where players need it, so a constant is a genuinely good approximation there.

The maximum is exact and worth knowing: setting the derivative to zero gives $\cot^2\varphi = 3.5$, so the peak is at $\varphi = 28.13°$ and the deflection there is $33.75°$. Nothing can deflect a rolling cue ball further than 33.75° from its original line. If you have planned position that needs 40°, you are not going to get it with a rolling cue ball at any cut angle, and no amount of stroke will help. That ceiling is the honest fact the rule of thumb hides.

And here is where the rule fails. Both ends of the curve collapse to zero. On a nearly full hit the cue ball barely deviates — it stops, then follows through nearly straight. On a very thin cut it barely deviates either, because it hardly interacted. At 60° the deflection is already down to 20.6°, at 75° it is 11°, and a player using "about 30°" on a thin cut will find their cue ball a foot or more from where they planned. Thin cuts are exactly where the rule of thumb should be abandoned, and exactly where beginners keep applying it.

Three assumptions, all of which can bite.

Full natural roll on arrival. If the object ball is close, the cue ball may still be sliding when it gets there, and the answer lands somewhere between this curve and the 90° stun answer. Use the slide-distance page to check.

The collision changes no spin. It is close but not exact: the brief friction between the two balls does perturb the cue ball's rotation slightly, and any sidespin survives and will interact with the cloth afterwards. The model here is a no-english model.

Throw is ignored. The object ball is not going exactly along the line of centres either, so the whole picture is a degree or two soft. That does not move the plateau, but it does mean neither ball goes precisely where the geometry says.

Note finally that the deflection ANGLE is independent of speed, but the DISTANCE the cue ball travels along that new line is not, and neither is the curved transition between the tangent line and the final direction. The ball does not turn a corner; it arcs. This formula gives the direction it ends up going, not the path it takes getting there.