The physics of a breaking putt.

1. Introduction

Reading a putt means answering three questions: which way it breaks, by how much, and how hard to hit it. Golfers answer them by eye, and they answer the second one badly. Dave Pelz's tests found golfers of every level predicted about a quarter of the true break [4]. Alpenfels and Christina found 96% of golfers under-read their putts by more than 60% [5]. The physics of the question, by contrast, is well understood: Holmes worked out how a ball and hole interact [2], and Penner modelled the path of a ball on a sloped green and the launch conditions that hole it [1].

This note sets out the model we use in Plumb Bob, a green-reading app for iPhone, in a form a technically minded golfer can check. We give the equations, solve typical putts, and draw out the results that matter for reading greens by eye: how break scales, where on the putt it happens, how speed and slope change the stroke, and why the oldest green-reading method under-reads. The figures and every number in this note are generated by the same simulator, at build time.

2. The model

2.1 A ball rolling on a slope

Describe the green as a height field h(x, y). A ball rolling without slipping has kinetic energy ½mv² + ½Iω² with ω = v/r. For a uniform sphere, I = ⅖mr², so the gravitational pull along the slope is reduced by the factor 1/(1 + ⅖) = 5/7. Adding rolling resistance as a constant deceleration μ opposing the velocity gives the horizontal equation of motion:

The first term pulls the ball down the fall line. The second slows it along its direction of travel. For the slopes found on greens (|∇h| ≲ 0.05) the square root is within 0.2% of 1. Note that the resistance acts on the ball's whole velocity, including its sideways component. That detail matters: it damps the sideways drift, and a common back-of-envelope estimate that ignores it puts the 10-foot aim at 14.7 inches instead of 11.0 (Section 4.2).

2.2 Green speed sets the rolling resistance

A Stimpmeter releases a ball at v₀ ≈ 1.83 m/s (6 ft/s) onto a flat part of the green, and the green's speed S is how far it rolls, in feet. On flat ground the ball decelerates uniformly at μ, so S = v₀²/2μ. Inverting gives the one parameter the model needs:

Table 1. Rolling resistance for common green speeds.

For comparison, a 2% slope contributes (5/7)·g·0.02 = 0.14 m/s² along the fall line: about a quarter of the rolling resistance at Stimp 10 (0.55 m/s²). Gravity is never the dominant force on a putt, but it acts for the whole roll, and near the hole, where the ball is slow, it is enough to turn it.

2.3 Solving a putt

We integrate the equation of motion with explicit Euler steps of 2 ms until the ball's speed falls below 5 mm/s. To find the putt that holes, we use a shooting method with two nested bisections: the launch speed is chosen so the ball would come to rest 17 inches (0.43 m) past the hole, Pelz's recommended pace [4], and the launch angle so that the path crosses the hole's centre. The two are alternated until both converge. We report the result as the aim offset: how far outside the hole, measured at the hole's distance, the ball must start. All results below are for a plane of constant slope. Plumb Bob applies the same solver to a measured, curved surface.

3. Measuring green speed

3.1 A stopwatch is enough

Because any rolling ball decelerates at μ, the ramp only standardises the launch. Time a ball from a moment it is truly rolling until it stops. Over distance d in time t it decelerated at μ = 2d/t², so

A ball that rolls 10 feet in 3.3 s gives a Stimp of 9.8. Since S ∝ t², a 3% timing error becomes a 6% error in Stimp. The practical procedure is in how to measure green speed without a Stimpmeter.

3.2 The Brede correction is exact

On a grade, gravity adds γ = (5/7)g sin θ to the deceleration rolling up and subtracts it rolling down, so S_up = v₀²/2(μ + γ) and S_down = v₀²/2(μ − γ). Brede's formula [3], recommended by the USGA for sloped greens, is their harmonic mean, and the slope cancels exactly:

The simple average of the two directions, by contrast, always overstates the speed on a slope.

4. Results

Unless stated otherwise: a 10-foot putt, a 2% side slope falling to the right, flat along the line, Stimp 10, pace 17 inches past.

4.1 The shape of a breaking putt

The path is not an arc. For most of the roll the ball is fast and goes nearly straight along its start line; the curve concentrates at the end.

4.2 Break grows faster than distance

Table 2. Break against putt length (2% side slope, Stimp 10). One cup = 4¼ in.

Doubling the length from 10 to 20 feet multiplies the break by 2.29, and a 30-footer needs 3.3 feet of break. A simple estimate helps explain the shape. If the sideways pull γ acted alone on a ball that takes time T to reach the hole, the offset would be ½γT², and on flat ground T = √(2/μ)·(√(D + p) − √p), where p is the pace distance past the hole. That grows faster than D, as observed, but it gives 14.7 inches for the 10-footer, overstating the simulated 11.0 because it ignores the rolling resistance acting on the sideways velocity.

4.3 Faster greens break more

A faster green means less rolling resistance, so the ball is launched more gently and spends longer on the slope. From Stimp 8 to Stimp 12, the same putt needs 52% more break.

4.4 Downhill putts break more

The side slope is identical in every case, yet the downhill 2% putt needs 15.2 inches and the uphill one 8.6: 1.78×. Downhill, gravity sustains the roll, the ball is struck more softly, and it spends longer crossing the slope. This is the putt golfers under-read most.

4.5 Pace changes the line

Table 3. The same putt struck at different paces.

A putt that would die 6 inches past needs 13.2 inches of break; one that would run 4 feet past needs 7.8. A read is only correct for the pace it was made for. At the 17-inch pace the ball reaches the hole at 0.68 m/s, far below the ≈1.63 m/s Holmes found to be the fastest a ball can enter the centre of the hole and still drop [2]. A firmer pace gives up little in capture on centred putts, but it narrows the effective width of the hole for putts that catch the edge. Holmes and Penner treat this in detail [1, 2].

4.6 Where the break happens

Only about 35% of a 10-footer's sideways movement has happened when it is two-thirds of the way to the hole; the remaining 65% happens in the last third. For a 30-footer the final third holds 67%. The sideways displacement grows roughly with the square of time on the slope, and the ball spends a disproportionate share of its time near the hole, where it is slow. The practical conclusion: read the last third of the putt hardest.

4.7 Plays-like distance

On a straight putt with grade γ (positive uphill), the launch speed that rolls a distance D and a further p past satisfies v² = 2(μ + γ)(D + p). The flat putt that needs the same stroke has v² = 2μ(D′ + p), so:

At Stimp 10, each 1% of grade changes a 10-footer's plays-like length by about 15%. A 2% uphill 10-footer plays 12.9 feet and a 2% downhill one 7.1. On fast greens the effect grows, because γ/μ grows with S: a 3% downhill 10-footer at Stimp 12 plays 4.8 feet. The golfer's version is in how hard to hit a putt.

4.8 The make zone

A ball drops if it can fall its own radius before reaching the far edge of the hole. For a ball of radius r crossing a hole of radius R with its path a distance b from the centre, that gives a capture speed of v_c = 2R√(g/2r)·√(1 − (b/R)²). For a centred putt this is 1.64 m/s, close to the 1.63 m/s Holmes derived from a full model of the ball–rim collision [2]. Figure 7 shows the result: the putts that drop form a curved band, not a point. There is no single correct line: every pace has its own. The band is widest, 5.5 inches of start line, for a ball dying 2 inches past, and narrows to 4 inches at the 17-inch pace and 3 inches at 4 feet past. A softer putt forgives the line but not the pace: leave it a few inches short and nothing drops. That trade-off is why the best pace depends on how well the golfer controls each (Section 9.2).

5. Plumb bobbing as geometry

To plumb bob, a golfer stands behind the ball, hangs the putter so the shaft is vertical, covers the ball with it using the dominant eye, and sees which side of the shaft the hole appears on. The shaft only marks a vertical plane through the eye and the ball; the information comes from the golfer's stance. Standing on a side slope s with the body perpendicular to the ground, the eye at height e is displaced downhill by about e·s. Sighting from there, a distance L behind the ball, the vertical plane through eye and ball passes the hole, at distance D, offset by

toward the low side, so the hole appears on the side the putt breaks toward, as the method says. With e = 1.6 m, L = 1.2 m and s = 2%, δ is 3.2 inches at 10 feet. The putt needs 11.0. The plumb bob is linear in D and independent of green speed and grade, while the true break grows as D^1.22, rises with Stimp and nearly doubles downhill (Figure 2). It gives the direction and badly underplays the amount. It also reads the slope at the golfer's feet, not along the ball's path, so a double breaker fools it entirely. More in how to plumb bob a putt.

6. Why golfers under-read

The model suggests three reasons, each consistent with the field studies [4, 5]. First, scaling: intuition expects break proportional to length, but it grows faster, so the error widens on longer putts. Second, hidden multipliers: speed and grade change break by 50–80% (Sections 4.3–4.4) without changing how the slope looks. Third, timing: most of the curve happens in the last third, the part of the putt that golfers who read from behind the ball see worst. Plumb bobbing, a common check, compounds the problem by showing only 23–36% of the break between 5 and 40 feet (Figure 2). The practical fixes for golfers are in why you miss putts on the low side.

7. A break formula for your head

We solved 5,070 putts spanning 3–40 feet, side slopes of 0.5–4%, grades from 3% downhill to 3% uphill and green speeds from Stimp 7 to 13, and fitted a power law to the aim offsets. The free fit gives aim ∝ s^1.02 D^1.23 S^1.08·e^-0.14·grade with a median error of 5%. Break is almost exactly proportional to slope and to green speed, and grows as length to the power 1.2. Rounding the exponents gives a rule a golfer can use on the green:

where grade is positive uphill. For a 10-footer it collapses to half the slope times the Stimp: 2% at Stimp 10 is about 10 inches (the full model says 11.0). For other lengths multiply by 0.4 at 5 feet, 2.4 at 20 feet and 3.9 at 30 feet. Downhill, add 12% per 1% of grade; uphill, take it off. Across all 5,070 putts the rule's median error is 6% (90% of putts within 16%); within 25 feet, 3% slope and 2% grade, it is 4% (90% within 12%). That is far smaller than the 60–75% by which golfers typically under-read (Section 6). The putt break calculator gives the full solution.

8. Grain: the slope you can't see

On grasses with strong grain, above all Bermuda, the blades lean in one direction and push a rolling ball that way. The simplest model treats grain as a small constant force along the grain direction, with magnitude γ_g. That is exactly the form of the slope term in the equation of motion, so in this model a ball cannot tell grain from slope. Grain is an invisible slope, with an equivalent size s_g = γ_g/((5/7)g). Three consequences follow.

A roll test measures slope and grain together, and the Brede correction removes both. Rolling into the grain and down it is identical to rolling up and down a slope, so the harmonic mean of Section 3.2 still returns the true green speed.

You can measure grain with two rolls. From the Stimp readings into and down the grain, S_into and S_down, the combined effect is

On a spot with no slope, that is all grain. On a sloped spot, subtract the slope you can see or measure along the roll: the rest is grain. A slope measurement, whether by eye, a level or an AR scan of the surface, cannot see grain; a roll can.

Table 4. Equivalent slope of grain from two rolls on flat ground (Brede speed about 9.9), and the break it adds to a putt across the grain at that speed.

A 2-foot difference between the two directions, from Stimp 9 into the grain to 11 down it, is grain worth 0.79% of slope. That alone breaks a flat 10-footer across the grain by 4.3 inches: a cup of break with no visible slope. It also explains why grain "trips up" golfers: it adds to or cancels the slope they can see. On a 1.5% side slope with that grain running against the slope, the net is 0.7% and the putt breaks 3.8 inches; with the grain running the same way as the slope, the net is 2.3% and it breaks 12.7 inches. Same visible slope, three times the break.

Two cautions. We have not measured grain strength; the table shows what a given roll difference implies, not how strong grain usually is. And real grain likely pushes harder as the ball slows and sinks into the blades, which would move its effect even further toward the hole than slope's (Section 4.6). Its direction follows the way the grass grows. Down-grain grass looks shiny and into-grain grass looks dull, and on Bermuda the grain often runs toward the setting sun or toward drainage, though those are rules of thumb rather than physics.

9. Simulating golfers

9.1 A golfer calibrated to the PGA Tour

To ask which errors matter, we simulate a golfer who reads the putt, plans the stroke for what they read, and then hits it imperfectly. Five sources of error are drawn from normal distributions on every putt: the slope read (standard deviation 25% of the true slope at skill 1), the grade read (0.5%), the green-speed estimate (1 Stimp), the start line (1°) and the launch speed (6%). The golfer's planned line and speed respond to their misread exactly as the solver's would. The ratios between these errors are assumptions; one overall skill factor k scales them all.

We fitted k so that the simulated golfer's average putts to hole out matches the PGA Tour benchmark from Broadie's ShotLink analysis [7] at every distance from 2 to 40 feet. Each distance was played on a mix of greens (side slopes 0.5–2.5%, grades ±1%, Stimp 11), and a missed putt's leave was scored with the model's own curve. The fit, k = 0.82, reproduces the benchmark with a root-mean-square error of 0.045 putts (Figure 8), and the simulated golfer holes 41% of 10-footers. It is slightly worse than tour players from 30–40 feet, where real lag putting beats the model's golfer by about a tenth of a stroke. The model has no bumps, spike marks or lip-outs from the rim's shape, so its golfer stands in for all of that too.

9.2 Is 17 inches past the right pace?

Pelz's 17 inches [4] came from real greens, where a firmer ball rides through the trampled ground around the hole. Our greens are perfectly smooth, so any preference for pace here comes only from the trade-off in Figure 7 and from the length of the comebacker. For each length and grade we simulated 5,000 putts at each of ten target paces and scored the expected putts to hole out:

Table 5. Pace (inches past the hole) that minimises expected putts, 2% side slope, Stimp 11.

The best pace depends on length. From 4 to 8 feet it is about 36 inches past: firm enough to take the low-side misses out, with a comebacker the golfer rarely misses. Around 12–20 feet it is 17–24 inches, close to Pelz's number. From 30 feet it falls to 9 inches, because the chance of holing is small and a long comebacker is costly. Dying a short putt at the hole is the most expensive mistake in Figure 9: at 4 feet it costs 52.3 strokes per 100 putts. Grade moves the optimum less than length does (Table 5); if anything, short downhill putts want a little more pace, because a firm putt breaks less on the slope that punishes a misread most.

9.3 What costs strokes: an error budget

Starting from the calibrated golfer on a slightly downhill 2% slope, we removed one source of error at a time and measured the strokes saved. The table is the answer to "if I could fix one thing":

Table 6. Strokes saved per 100 putts by removing one source of error, by putt length. The largest in each column is highlighted.

Three patterns stand out. Reading the slope is the biggest single cost from 5 to 10 feet for this golfer; beyond that, pace takes over. Start-line error matters mostly inside about 5 feet. The putting stroke gets most of the attention in practice, but past tap-in range reading and pace dominate. Misjudging the green speed adds a steady cost at every length, and grows on long putts, where it moves both the pace and the break.

9.4 Under-reading, over-reading and the low side

Finally we gave the calibrated golfer a systematic read bias: reading 25% to 175% of the true slope, with everything else unchanged.

The cost curve is close to symmetric: at 6 feet, reading 75% of the break costs 15.9 strokes per 100 putts and reading 125% costs 11.6. Missing low is not much worse than missing high for the same size of misread, although low misses do leave longer comebackers (1.8 against 1.6 feet at 6 feet). The low side is where golfers miss because the typical misread is so large and so one-sided. Reading 25% of the break, the level Pelz reported [4], costs 56.8 strokes per 100 six-footers; Pelz's advice to triple the read brings the golfer to 75% and cuts that to 15.9.

10. Limitations

• Planar greens. Results are for constant slopes. Real greens curve; the app solves on a fitted surface, but the scaling results here are for planes.
• Constant rolling resistance. Real deceleration varies with speed and grass (it rises as the ball slows and settles into the grass). The Stimp calibration fixes the average, not the shape.
• No wind, moisture or skid. The initial skid after impact is ignored.
• Simple hole capture. The capture rule in Section 4.8 ignores the rim's shape and lip-outs; see Holmes [2] and Penner [1] for a full treatment.
• Simulated golfers. The relative sizes of the five error sources are assumptions; only their overall scale is fitted. Real golfers differ, and their errors are correlated. Treat Section 9 as showing mechanisms and orders of magnitude, not a prescription for any one golfer.
• Grain is modelled as a constant force; the strength of real grain is not measured here.
• Plumb-bob geometry assumes a golfer standing perpendicular to the slope; a golfer who stands plumb-vertical sees nothing.

11. From model to app

Plumb Bob applies this model to a real green. As the golfer walks from the ball to the hole, the iPhone's AR tracking (and LiDAR, on Pro models) samples the surface; the app fits a height field, measures the green speed from one rolled ball using the method of Section 3, and runs the solver of Section 2.3. It then draws the line, the aim point in inches, the break in cups and the plays-like distance on the grass. For a planar green you can run the model yourself in the putt break calculator. The plumb step uses the phone's gravity sensor for a true vertical. A device read is not allowed in competition under Rule 4.3 of the Rules of Golf; see are green reading apps legal?

References

1. Penner, A. R. (2002). The physics of putting. Canadian Journal of Physics, 80(2), 83–96. doi:10.1139/p01-137
2. Holmes, B. W. (1991). Putting: How a golf ball and hole interact. American Journal of Physics, 59(2), 129–136. AIP
3. Brede, A. D. (1990). Measuring green speed on sloped putting greens. USGA Green Section Record. PDF
4. Pelz, D. (2000). Dave Pelz's Putting Bible. Doubleday.
5. Alpenfels, E. and Christina, R. Evidence-Based Golf. Summarised in GOLF.com.
6. R&A and USGA. Rules of Golf, Rule 4.3. randa.org
7. Broadie, M. (2011). Putts gained: measuring putting on the PGA TOUR. Columbia University. PDF

Parameters

Cite as: Plumb Bob (2026). The physics of a breaking putt. Technical note, version 1.1. https://plumbbob.app/research/putt-physics/