The run moves the best drive below the ridge that carries furthest

A drive’s best spin falls where players hit it gave a golf ball the lift and drag coefficients wind tunnels measure on spinning balls — lift growing as the spin ratio to the 0.4 power, drag rising with it — and found that the carry of a 70 m/s drive has a best spin, 2,273 rpm, with a best launch of 16.8°, for 234.6 metres. It found the maximum shallow in one direction. More spin and a lower launch, or less spin and a higher one, carry almost as far, along a ridge on which the launch falls from 20.4° at 1,500 rpm to 10.7° at 4,000 while the carry stays within three metres. And it named what its model stopped at: the first landing. On a firm fairway the carry is a drive’s larger part, not the whole of it.

Its question was where along that ridge the total is greatest. A lower launch with more spin was said to land more shallowly and run further; its extra backspin brakes the ball at the bounce. Which wins decides whether the run moves the best launch from 17° towards the eleven to fourteen degrees launch monitors recommend.

The answer is that neither wins, because the premise fails. Every drive on the ridge lands at almost the same angle, so the shallower landing the low launch was supposed to buy is not there, and the run hardly chooses between them. What the run rewards is leaving the ridge altogether — less spin and a lower launch — and on an ordinary fairway it moves the best drive in all to 1,359 rpm and 11.6°, landing near thirty-five degrees.

A bounce off the wall of its own crater #

The flight is the earlier essay’s, unchanged: the ball’s 45.9 grams and 42.7 millimetres, sea-level air, drag and lift from the spin ratio $S = \omega R/V$ through Smits and Smith’s fitted forms, and the spin held constant over the flight. The dimples keep the ball past the drag crisis for the whole of it, so the coefficients are the supercritical ones throughout. What is new is the ground.

A ball landing at thirty metres a second on turf is not a ball bouncing off a floor. The turf yields, the ball presses a crater, and by the time the impact is at its hardest the ball is striking the crater’s front wall, a surface tilted up towards it. Penner’s study of a golf ball’s run, in 2002, made that tilt the central thing and gave it a form: it grows in proportion to the impact speed and to the steepness of the landing. The same form is used here, with the tilt 10° for a landing at 19 m/s and 44° on a medium fairway, and the bounce is computed in the tilted frame and turned back to the level.

The bounce itself is the textbook rigid-sphere impact. The ball’s normal speed into the surface comes back multiplied by a coefficient of restitution, 0.12 for turf, which keeps very little. Along the surface the ball’s contact point is slipping forward — at its forward speed plus its backspin’s surface speed, since backspin turns the bottom of the ball forward — and friction acts against the slip with an impulse that can be at most the friction coefficient times the normal impulse. If that is enough to stop the slip, the ball leaves the surface rolling; if not, it leaves still sliding. Each hop is then a short ballistic arc, and when the rebound is too small to call a hop the ball rolls, decelerating at a constant 4 m/s².

That deceleration and the turf’s other numbers are stated, not measured. They were set so that one drive of a familiar shape — 2,700 rpm at 11°, landing at 42° — runs about twenty metres, which is the order such a drive is reported to run on an ordinary fairway. A firm fairway, with half the crater and a livelier roll, and a soft one, with more of both, bracket it. The argument below is about how the run depends on the launch, and that dependence comes from the impact’s mechanics, not from the calibration; the calibration sets only how long the runs are.

The ridge of best carries runs almost flat #

The first figure puts the run onto the ridge. At each spin rate the drive is launched at the angle that carries it furthest, so the dashed curve is the carry the earlier essay drew; above it are the totals on three fairways. All three are nearly as flat as the carry. On the medium fairway the total peaks at 2,052 rpm, 221 rpm below the carry’s best, and the whole ridge from 1,200 to 3,200 rpm lies within two and a half metres of it. On the soft one it peaks at 2,375 rpm — slightly more spin than the carry wants. Only the firm fairway moves the best spin by much, to 1,728 rpm, and even there the gain over the carry’s own best launch is under a metre.

The reason is in the last line of the figure’s own numbers: every drive on the ridge lands between 45° and 49°. A launch at 23° with 1,000 rpm and a launch at under 10° with 4,400 rpm land within four degrees of each other. They land at almost the same speed too, 27 to 31 m/s. And the run of a ball depends overwhelmingly on how steeply and how fast it lands, as the figures below show. Two drives that arrive alike run alike, and what is left to separate them — the backspin at the first bounce — moves the best spin by a couple of hundred rpm.

The ridge holds the landing angle fixed #

The second figure shows why the shallow landing never arrives. At any one spin rate, the landing angle rises steeply with the launch: at 2,300 rpm a drive launched at 8° lands at 35°, one launched at 20° lands at 50°. But the ridge does not move along a line of fixed spin. Going down the ridge, the launch falls and the spin rises, and the two effects on the landing cancel almost exactly: the lower launch would flatten the descent, the extra lift steepens it again, and the ridge’s landing angle falls by only four degrees while its launch falls by fourteen.

The cancellation is computed rather than derived, and it is not that the flights are the same shape. The apex falls along the ridge, from 50 metres at 1,000 rpm to 40 at 4,400, so the spinnier drive flies a lower, longer arc. A spinning ball should leave low found that lift on a descent flattens it and drag steepens it, and the measured curves decide which wins. By the descent the ball has slowed and the spin has not, so the spin ratio is at its highest — and there the measured lift has nearly stopped growing while the measured drag keeps rising. Fly the 4,400 rpm drive with lift in proportion to spin and it lands at 36°, ten degrees flatter than the 1,000 rpm drive; with lift that saturates it lands at 43°, and with drag rising too, at 45°. The curvatures that gave the carry its best spin are the same ones that hold the ridge’s landings together.

Off the ridge the flight changes shape. The curves of fixed spin cross the ridge steeply: lowering the launch with the spin held lowers the apex, and a lower apex at the same speed lands flatter. That is where a shallower landing comes from, and it costs carry, because the drive is no longer launched at its best angle for its spin.

At one spin the run rewards a lower launch, and stops #

The third figure prices it. At 2,300 rpm on the medium fairway the carry, dashed, is greatest near 17° and falls away below it — 225 metres at 8°, ten short. The total is greatest at 9°, 254.7 metres against 244.2 at 20°. Over that range of launches the run grows faster than the carry shrinks, because the landing is flattening and the run grows steeply with a flatter landing. Below 9° the carry starts to fall faster than the run grows, and the total turns down.

The same shape holds at 1,600 rpm, whose total is greatest at 11°, and at 3,200 rpm, at 7°. Each spin has a best launch for total distance, several degrees below its best launch for carry; less spin wants a higher launch, for the reason the carry’s ridge runs the way it does.

The dial changes the fairway. On the soft one the shape is the same, with the best launches up to a degree lower and the totals ten metres shorter: the deeper crater takes more of every landing, so the run is smaller and there is less to be gained by flattening it. On the firm fairway the shape is different. There the total keeps rising as the launch falls, all the way to the edge of the figure, and its best is not inside the range drawn: a drive launched at 4° with 1,600 rpm is credited 317 metres.

That number is the model failing, not a recommendation. A drive launched that low lands at under thirty degrees, and below that the ball is being skimmed across the ground rather than dropped onto it — the impact is glancing, the crater the turf model gives it is nearly flat, and a ball running at twenty metres a second through grass is not decelerated at the rate a slow roll is. Nothing in the turf’s stated numbers was set against such landings. The shaded band marks the launches at 1,600 rpm that land flatter than thirty degrees, and on a firm fairway the run’s preference lies entirely inside it. What the model says about firm ground is only that the best launch there is lower than on ordinary ground, and how much lower it cannot say.

The total has a ridge of its own #

Doing that at every spin rate draws a second ridge, the fourth figure: for each spin, the launch that goes furthest in all. It lies seven or eight degrees below the carry’s — at 2,200 rpm, 9.5° on the medium fairway and 8.9° on the soft, against the carry’s 17.1° — and it runs parallel to it, falling as the spin rises.

Unlike the carry’s ridge, this one has a clear best point. Along it the total peaks on the medium fairway at 1,359 rpm and 11.6°, 256.5 metres in all, and on the soft one at 1,779 rpm and 9.9°, 245.5 metres. The best carry, by comparison, is 2,273 rpm at 16.8°, and its drive finishes at 249.2 metres on the medium fairway. Seven metres separate the two — small against a fairway’s width, large against the differences fitting sessions argue over, and entirely the run’s.

Both best points land at nearly the same angle — 35.5° on the medium fairway and 35.4° on the soft — although they differ by 400 rpm and two degrees of launch. That is the result in its plainest form. The run cares about the landing, and the best drive in all is the one whose flight is shaped to land at the angle at which the last few metres of carry given up and the run gained are traded evenly. On these two turfs that angle is about thirty-five degrees.

The crater decides what survives the bounce #

The fifth figure takes the first impact apart, for a drive at 2,300 rpm launched anywhere from 6° to 24°. A drive landing at 31° arrives with 23 m/s of forward speed and leaves the first bounce with 12.4; one landing at 52° arrives with 19 and leaves with 4. The arriving speeds differ by a fifth. The leaving speeds differ by a factor of three.

Most of that factor is the crater. A steep, fast landing presses a deep crater, and its front wall is tilted further towards the ball, so more of the ball’s forward motion is driven into the turf as a normal impact — which the turf’s restitution then keeps only an eighth of. On a turf that did not yield, the dashed curve, the same drives would leave at 16 and 12 m/s: an impact that is mostly friction, which slows a steep landing more than a shallow one but not by much. The yielding turf turns the landing angle into the main thing.

The spin’s share is the gap between the solid curve and the curve with the spin removed. Backspin turns the bottom of the ball forward, so the contact point slips faster than the ball moves, and friction has more slip to stop; the ball leaves rolling, but rolling slower. At a shallow landing the spin costs 1.5 m/s of the 13.9 the impact would otherwise leave; at a steep one it costs 1.4 of 5.4, a quarter of what is left. That is the braking the earlier essay’s question had in mind, and it is real — on the carry’s ridge, where the landing is fixed, it is the only thing left to decide between drives, which is why the ridge’s best total moves towards less spin. But it is the second effect, not the first.

The impact also respects the conservation it should. Through an impact that ends with the ball rolling, the ball’s angular momentum about the contact point cannot change, since friction and the normal force both act through that point; the computed bounces keep it to rounding error, and a ball landing already rolling feels no friction at all. That is the mechanics the bounce in a ball that bounces in water and not in oil does not need — there the bounce is decided by a film, here by a solid — and both come down to the same question of what the surface can take out of the ball before it leaves.

Three drives on the same fairway #

The sixth figure draws the three drives that matter. The best carry rises to 46 metres, lands at 47° after 234.6 metres and runs 14.6. The tour-shaped drive — the one the turf was calibrated against — rises to 35 metres, lands at 42° after 232.5 and runs 19.4, for 251.9 in all. The best drive in all rises to only 26 metres, lands at 35.5° after 221.5 and runs 35, for 256.5.

Read from left to right, the figure is the trade. Each step down in apex costs a few metres of carry and returns more than that in run, until the flight is low enough that the carry’s losses outrun the run’s gains. The best drive gives up thirteen metres of carry to the run and gets back twenty. To a spectator it looks unimpressive — a lower, flatter ball that seems to have been under-hit — and it finishes furthest down the fairway.

The best drive’s run comes in two nearly equal parts. Its first hop, off the crater’s wall, carries it another thirteen metres, the next two four and one; then it rolls, for eighteen. The hops are set by the impact and the roll by the turf’s deceleration, which is the number every metre of the roll is divided by and the one a fairway’s firmness changes most from week to week. It is why the same drive can run twenty metres in a dry August and five in a wet March without anything about its flight being different.

Faster balls want less of both, and land the same #

The seventh figure repeats the search at four ball speeds. For carry, the earlier essay’s result: faster balls want less spin and a lower launch, from 2,837 rpm and 22.7° at 50 m/s to 2,022 rpm and 14.9° at 80 — the same direction the terminal speed sets the best launch angle found for drag alone, since a faster ball is further above the speed at which drag and weight balance. For total distance on the medium fairway the best spin is much lower and almost independent of speed — 1,159 to 1,359 rpm — and the best launch falls from 17.9° to 9.2°, about five degrees below the carry’s at every speed. On the soft fairway the best spin is a few hundred rpm higher, again nearly constant.

The best drive’s landing angle hardly moves with ball speed either: 32.8° at 50 m/s, 34.1° at 60, 35.5° at 70, 36.9° at 80, on the medium fairway. A slow swing and a fast one, launched at very different angles, are best shaped to land at about the same angle. That is the closest thing this calculation has to a rule a player could carry onto a course: on ordinary ground, a drive is best flown to land at about thirty-five degrees, and the launch and spin that achieve that are whatever the ball speed makes them.

What it says about the launch monitors #

The earlier essay found the carry’s best launch, 16.8°, a few degrees above the launch monitors’ eleven to fourteen, and named roll as the first thing that might bring it down. It does. The best total on the medium fairway launches at 11.6° at 70 m/s and lands at 35.5°; recommended launch conditions for drives of this speed sit in the same range, and tour drives are reported to launch near eleven degrees and land in the high thirties, close to the shape the model calls best.

What does not match is the spin. The model’s best total wants about 1,400 rpm; real drives at this speed carry 2,000 to 2,700, and that is what monitors recommend. The gap is the club, not the air. The model chooses launch and spin independently, but a driver does not deliver them independently: both come from the same loft and the same angle of attack at impact, and lowering the spin far below 2,000 rpm with a driver also changes the launch and the strike’s consistency. A ball spinning at 1,400 rpm also becomes sensitive to the small changes in its coefficients near low spin ratios — the region where the ball that swings without spinning finds a ball’s flight least predictable — and a drive that cannot be repeated is not an optimum anyone can use. So the model’s result is better read as a direction than a target: from the carry’s best, the run rewards less spin and a lower launch, together, until the landing is near thirty-five degrees.

The convention: carry, run and total #

Carry is the horizontal distance to first landing on level ground in still sea-level air, as before. The run is everything after: the hops, any skid, and the roll, on level turf. The total is their sum, measured along the line of flight. Backspin is about a horizontal axis and is held over the flight; at the first impact it is the flight’s spin, and from then on the impulses change it. The turf’s numbers are those of a model fairway, stated in the method, and “medium” means the one set against a tour-shaped drive’s twenty-metre run. Best means greatest total, found by golden-section search over the launch at each spin, and then over the spin.

What the ground model leaves out #

The turf’s numbers are not measurements. A coefficient of restitution of 0.12, a friction coefficient of 0.45, a crater tilt of 10° for a reference landing and a roll deceleration of 4 m/s² describe a plausible fairway, set by one landmark. A real fairway’s numbers depend on the grass, its cut, the soil and its moisture, and they change through a season.

The roll is decelerated at a constant rate. That is the Stimpmeter’s description of a green, measured at walking pace. A ball running fast through longer grass is slowed harder, and a constant rate credits a fast roll with too much. It is the reason the model’s firm fairway runs away and its results there are not reported as answers.

Flat landings are outside it. Below about thirty degrees the ball skims, and nothing here was set against a skimming impact.

The ground is level and uniform. Real fairways slope, and a drive landing on a down-slope runs further and one landing on an up-slope stops. The bounce’s direction is assumed to stay along the line of flight; real bounces scatter sideways, more so the further the ball runs.

The flight is the earlier essay’s, with its spin held constant, the coefficients one ball’s, and no wind. The ball that never forgets its spin found the spin’s decay over a drive to be a few per cent, and its effect on the landing angle is smaller than the differences drawn here.

Who studied the run #

Golf balls’ impacts with turf were measured with high-speed cameras in the 1990s, by Haake among others, and Penner’s analysis of a ball’s bounce and roll, including the crater whose front wall the ball meets, was published in 2002. Launch monitors, which tracked drives by radar or camera from the early 2000s, made total distance a measured quantity for every strike, and their manufacturers’ optimisation tables for total distance are built on turf models of their own. The finding here that the carry’s ridge holds the landing angle almost fixed, and that the best total lands near thirty-five degrees, is this calculation’s, from the model above; it inherits every assumption of the turf model and of the spin-dependent flight.

How the parts were checked #

The impulse was checked three ways. A ball that lands already rolling, its contact point at rest, must feel no tangential impulse, and feels none. A ball whose slip is stopped within the impact must leave at the closed-form rolling speed, its forward speed less two-fifths of its spin’s surface speed, divided by seven-fifths, and it does to rounding error. And angular momentum about the contact point is unchanged by an impact that ends rolling. Halving the flight’s time step moves the total by two micrometres. The ledger also records the calibration — the tour-shaped drive’s landing at 41.8° and run of 19.4 metres — and the two best points.

Still open: a driver that sets launch and spin together #

The model treated launch and spin as two free numbers, and its best drive in all asked for less spin than a driver usually gives. A real strike sets both from the same two things — the loft the club presents at impact and the direction it is travelling — so the pairs a player can reach are a line across the plane drawn here, not the whole plane, and the line moves with the club. The next calculation would replace the free pair with an impact model of the club on the ball, launch and spin as functions of loft and angle of attack, and ask where along that line the total is greatest: whether a player gains more by changing the driver’s loft or by changing the angle of attack, and whether the thirty-five-degree landing survives when spin cannot be chosen on its own.