Is it better to "hit it firm" or to "play the line"?
- Distance & pace
You are told to "get it past the hole", and also to "use the break with a dying putt". Which is right? Looking at the physics of the cup itself, the answer is clear.
In this article we start from the speed at which a cup can still catch a ball, and work out in numbers how the aim window and the break change as you change the pace. All numbers come from the same physics model as the Putt Break Calculator.
The map above is the calculator's view of a 5 m putt on a 1° side slope: how likely each combination of pace and aim is to go in. Inside the white dashed outline the putt always drops, and that zone is a thin band running diagonally from top left to bottom right. Hit it firmer and you aim closer to the hole; hit it softer and you aim further out. Pace and aim cannot be chosen separately — you can see that at a glance. The rest of this article works out where in that band you are most likely to hole it.
The cup rejects a ball that is going too fast
Once the ball crosses the near edge of the cup it loses its support and starts to fall. If it cannot drop by one ball radius before reaching the far edge, it hits the rim and runs on out.
A ball passing a distance d from the centre of the cup has to cross 2√(R² − d²) of open hole (R = cup radius, 54 mm). At speed v that takes 2√(R² − d²) / v, and in that time it falls ½ g t². Requiring it to fall one ball radius r (21.3 mm) gives
½ g (2√(R² − d²) / v)² ≥ r
⇒ v ≤ v_max(d) = 2√(R² − d²) · √(g / 2r)
Through the middle (d = 0) this is about 1.63 m/s, matching the critical speed given by Holmes (1991). Any faster and the ball simply flies over the hole.
The further from the centre, the shorter the crossing, so the allowed speed drops.
| Offset from centre d | 0 cm | 1 cm | 2 cm | 3 cm | 4 cm | 5 cm |
|---|---|---|---|---|---|---|
| Maximum speed that drops | 1.64 m/s | 1.61 m/s | 1.52 m/s | 1.36 m/s | 1.10 m/s | 0.62 m/s |
The cup has a 5.4 cm radius, but when the ball's centre passes within 1 cm of the edge (d = 4–5 cm), it has to be going at around 1 m/s or less to drop. In other words, the firmer you hit it, the narrower the cup effectively becomes.
Too soft and you cannot read the break
Go the other way and the ball breaks sharply at the end. How tightly the path bends (its curvature) is the sideways acceleration divided by the square of the speed:
curvature κ = a_side / v²
On the same side slope, halving the speed makes the curvature four times larger. The ball slows as it nears the hole, so the same slope bends it more late in the putt. The softer you hit it, the more of that late break you have to allow for. (Which part of the slope should weigh most when you pick your aim is a separate question, covered in "Is AimPoint physically correct?". The slope in the last metre before the hole affects the aim least.) And of course, a putt that never reaches the hole goes in 0% of the time.
What changes when you change the pace
Take a 3 m putt with a 1° side slope at Stimp 10. We varied how far past the hole the ball would finish (the overshoot), and for each pace computed the aim that sends the ball through the middle of the cup and the range of aims around it that still hole out. The "aim window" is that range expressed as a width at the hole.
| Overshoot | Speed at the hole | Aim window | Break | Aim (beside the hole) |
|---|---|---|---|---|
| 10 cm | 0.40 m/s | 13.8 cm | 11.5 cm | 33 cm |
| 20 cm | 0.51 m/s | 13.3 cm | 10.5 cm | 31 cm |
| 30 cm | 0.60 m/s | 10.7 cm | 9.5 cm | 29 cm |
| 40 cm | 0.68 m/s | 10.3 cm | 8.7 cm | 27 cm |
| 60 cm | 0.81 m/s | 9.8 cm | 7.7 cm | 24 cm |
| 1 m | 1.01 m/s | 8.7 cm | 6.2 cm | 20 cm |
| 1.5 m | 1.22 m/s | 7.3 cm | 5.0 cm | 17 cm |
| 2 m | 1.40 m/s | 5.6 cm | 4.3 cm | 15 cm |
Three things stand out.
- The aim window narrows the firmer you hit it. From 13.8 cm at 10 cm past, it shrinks to 5.6 cm at 2 m past, about 40% of the original. A slow ball drops even when it just catches the edge, which is why the window can be wider than the cup's 10.8 cm diameter
- The break also shrinks the firmer you hit it. A line that breaks 11.5 cm at 10 cm past breaks only 6.2 cm at 1 m past. "Hit it firm and take the break out" is only half true
- The aim almost doubles. At 10 cm past you aim 33 cm right of the hole; at 2 m past, 15 cm. You cannot choose the aim without first choosing the pace
The key point is that once you choose the pace, there is exactly one correct aim. "Hit it firm" and "play the line" are not competing options. It is a question of order: decide the pace first, then start the ball on the line that matches that pace.
Allowing for pace error
On the table alone, a dying putt finishing 10–20 cm past looks best. But nobody hits the exact pace they intend. Aim to die it in the hole and a putt struck a little soft never gets there.
So we gave the launch some dispersion in pace and direction, and searched for the overshoot that maximises the chance of holing out. Pace error is treated as log-normal and direction error as normal, integrated over the "goes in" region above.
| Skill (pace / direction dispersion) | 3 m, flat | 3 m, 1° side slope | 5 m, 1° side slope |
|---|---|---|---|
| Advanced (4% / 0.8°) | 21 cm (79.5%) | 26 cm (75.2%) | 32 cm (49.4%) |
| Intermediate (8% / 1.5°) | 40 cm (46.3%) | 43 cm (44.2%) | 59 cm (25.2%) |
| Beginner (14% / 2.5°) | 62 cm (26.1%) | 62 cm (25.4%) | 80 cm (13.2%) |
Each value is the optimum overshoot, with the holing probability at that pace in brackets.
The larger the pace dispersion, the further the optimum moves towards running it past. Giving up a little of the aim window pays off because it avoids the certain miss of leaving it short. As a rule of thumb, that is 20–30 cm for an advanced player, 40–60 cm for an intermediate player and 60–80 cm for a beginner. Longer putts carry bigger pace errors, so the optimum moves further past the hole.
How slope changes this
All of this does shift with slope and green speed. On a steep downhill there is almost no room to run it past, so the optimum moves towards a dying putt. Downhill, the link between the speed at the hole and the overshoot is completely different from a flat putt. We test this in detail in "Is "17 inches past" really the best pace?".
Summary
- There is a maximum speed at which the cup will catch a ball: 1.63 m/s through the middle, dropping sharply towards the edge
- The firmer you hit it, the narrower the aim window and the smaller the break
- Choose the pace and there is exactly one correct aim. The pace is what you should be deciding
- Allowing for dispersion, the optimum overshoot is 20–30 cm for an advanced player and 40–60 cm for an intermediate player
- On a steep downhill, the optimum moves towards a dying putt
Look at the "map of what goes in" in the Putt Break Calculator and you can see the shape of those tolerances at a glance. Tall and narrow means a line that forgives pace; wide and flat means a line that forgives direction.
Conditions: this site's physics model. Stimp 10, launch angle 2°, 20% topspin at launch, 15% low-speed rise in resistance. "Break" is the largest distance by which the path the ball would take without a hole strays from the straight line between ball and hole.