A drive's best spin falls where players hit it only because lift tires and drag grows

A spinning ball should leave low gave a ball a lift as well as a drag and found that a glider should leave the ground at a low angle: a 70 m/s drive with 3,000 rpm of backspin, its lift coefficient 1.6 times the spin ratio and its drag coefficient fixed at 0.25, carries 262 metres at a best launch of 17.1°, against 202 metres at 38.1° without spin. It ended by naming the two things its model held fixed that a real ball does not. As a drive slows, the spin ratio $S = \omega R/V$ rises — the spin is nearly kept, the ball never forgets it, while the speed falls by half — and measured lift does not keep pace with $S$, while measured drag grows with it. And launch monitors, which fit real drives, recommend eleven to fourteen degrees rather than seventeen.

Putting the measured curves in does more than move the angle. It changes what kind of question the best spin is.

Two curves a wind tunnel measures #

Smits and Smith fitted wind-tunnel measurements of spinning golf balls at the Reynolds numbers of a drive to two simple forms: a lift coefficient $C_L = 0.54\,S^{0.4}$ and a drag coefficient $C_D = 0.24 + 0.18\,S$. Against the fixed pair they differ in shape more than in size. At the spin ratio a drive is launched with, about 0.07, the measured lift is larger than $1.6S$; by the time the ball has slowed to land, near 0.17, it is smaller. The measured lift bends over — each increment of spin buys less lift than the last — and the measured drag climbs, by a tenth over a flight.

Both curvatures have a reason in the boundary layer. Spin moves the separation points round the ball, one forward and one back, and the lift comes from the asymmetry; the asymmetry cannot grow without limit, because the separation points have only so far to go, so the lift saturates. And a spinning ball drags a wider wake behind it than a still one, so the drag rises. The drag that falls as it speeds up — the drag crisis that dimples are there to trigger — sets the drag’s level; the spin sets its slope.

The spin ratio a drive sweeps through #

The reason the curvature matters is that a drive does not sit at one spin ratio. The spin is nearly kept while the speed falls: the best measured drive leaves the club at 70 metres a second with a spin ratio of 0.073 and lands at 29 metres a second with one of 0.175, more than doubled. Every number that depends on the spin ratio is therefore a curve the ball travels along during its flight, and the shape of the curve, not its value at launch, decides what happens in the second half. The fixed pair and the measured pair can agree at launch and still fly differently, because they part company exactly where the ball spends the descending half of its flight.

Read as a lift-to-drag ratio, the difference is stark. With the fixed coefficients and 3,000 rpm the drive leaves at a lift-to-drag ratio of 0.61 and lands at 1.48: it becomes a better glider as it slows, which is what lets it float so far. With the measured coefficients the best drive leaves at 0.75 and lands at 0.99 — it barely improves, because the lift it gains from the rising spin ratio is matched by the drag it pays for it. A golf ball in the last third of its flight is a poor glider getting slightly less poor, and that is the regime in which the extra spin has to earn its keep. Lift without a wing found a spinning cylinder lifting exactly as its circulation says; a spinning ball’s circulation is set by separation points that can only move so far, and the measured lift is what that limit looks like from outside.

Without both curves the best spin is in the wrong place #

The flight is integrated in metres and seconds, gravity, drag along the velocity and lift across it, with the spin held over the flight and the carry — the distance to first landing — as the measure. With the fixed pair it reproduces the earlier calculation exactly: 202 metres at 38.1° without spin, 262 at 17.1° with 3,000 rpm. Then, at each spin rate, the launch angle that carries furthest is found, and the carries compared.

With the fixed pair, more spin carries further until the launch reaches level. The best carry rises from 211 metres at 500 rpm to 287 at 4,800, and all the while its best launch angle falls, from 35.5 degrees to nothing: spin is doing the climbing the launch used to do. There the optimum stops being a balance and becomes a wall. Past it the ball would need to leave the tee pointing at the ground, and held at level it balloons and carries less, 268 metres at 6,000 rpm and 248 at 7,000. Carried on past 8,000 rpm, the lift is so large that the ball climbs away on its own and the flight becomes a glide: 840 metres at fifteen thousand rpm, a lift-to-drag ratio that of a sailplane. That is not a golf ball; it is what a lift coefficient in proportion to spin with nothing to pay for it implies, and it says the fixed pair was usable only at a spin someone chose.

Swap in one measured curve at a time and the best spin moves down, but not far enough. With drag rising with spin and the lift still in proportion, extra spin costs drag while buying lift, and the carry peaks at 257 metres at 4,400 rpm with a launch of four degrees — a balance, but at nearly twice the spin of a real drive, and past 8,000 rpm the runaway returns. With the lift saturating and the drag fixed, extra spin buys less and less lift but costs nothing, and the carry creeps up across the whole range drawn, to 257 metres at 6,000 rpm, levelling near 260 only around 9,000 with the launch at level. Only both together put the maximum where drives are hit: 234.6 metres at 2,273 rpm and 16.8°, falling either side, with no runaway at any spin. A best spin exists near where drives are hit because the last revolution buys less lift than it costs drag at modest spin, and that needs a lift that tires as well as a drag that grows.

That is the surprising connection to the essay before: the terminal speed sets the best launch angle for a ball without lift, because drag alone has a cost and gravity a reward; for a spinning ball the same balance moves into the spin rate, and it needs the coefficients’ curvature rather than their size to have an interior answer where a drive is actually launched.

A ridge, not a peak #

The maximum is shallow along one direction. At each spin rate there is a best launch angle, and as the spin rises it falls: 20.4° at 1,500 rpm, 16.8° at the best spin, 10.7° at 4,000 rpm. Along that ridge the carry stays within 2.7 metres of its best. A drive launched at 13° with 2,700 rpm — a launch monitor’s kind of number — carries 233.9 metres, seven-tenths of a metre short of the best, and lands at 44° against the best’s 47°.

So the two numbers a fitter quotes are not equally firm. The spin is fixed by the coefficients’ curvature; the angle is a choice along a ridge, and a few degrees either way cost under a metre of carry provided the spin moves with it. Lower launch with more spin and higher launch with less spin carry the same; what separates them is everything after the first landing, which favours the shallower landing of the low launch.

Why the ridge runs the way it does #

Spin and launch angle are two ways of getting the same thing, height early in the flight. A higher launch puts the height in at the start, through the launch’s vertical speed, and pays for it with a lower horizontal speed; more spin buys height steadily through the flight with lift, and pays with the drag that comes with it. Near the optimum the two are exchangeable at a nearly constant rate, and the ridge is the line along which they are traded one for the other at no cost. That is also why a fitter can usually fix a drive that balloons or one that falls out of the sky by changing either number: the direction of the error matters more than which of the two corrects it.

The ridge is flat but not level, and it is steeper on the high-spin side. At its low-spin end, 1,500 rpm and 20°, the carry is nine-tenths of a metre below the best: too little lift, so the ball has to be thrown upwards and spends its flight at a poor angle to the air. At its high-spin end, 4,000 rpm and 11°, it is 2.7 metres below: too much drag in the second half, where the spin ratio is highest and every revolution is paid for at the drag’s steepest. Between them the best is broad, and a real drive that lands anywhere within a few hundred rpm of it carries within a metre.

Faster balls want less of both #

The best pair moves with ball speed in the direction fitting practice follows. At 50 m/s — a slow swing — the best drive spins at 2,837 rpm and leaves at 22.7°; at 70, 2,273 rpm and 16.8°; at 85, 1,911 rpm and 14.1°. A faster ball starts at a lower spin ratio for a given spin rate and keeps a lower one longer, so it needs less spin to reach the same lift, and the drag it would pay for more spin grows with the square of its speed.

How far the earlier model flew it #

The fixed coefficients’ best drive at 3,000 rpm carries 262 metres; the measured coefficients’ best, 235. The difference, 27 metres, is the lift the fixed pair credited late in the flight, where the spin ratio is high and the measured lift has saturated, and the drag it did not charge. The measured flights are lower in their second half and land more steeply, which is where a real drive’s shape departs most visibly from the parabola-with-a-glide the fixed pair draws.

A ball’s curves decide its best spin #

If the best spin is set by the curvature of the two coefficients, a ball with different curves has a different best spin, and the sensitivity is large. Keep everything else and let the lift grow as the spin ratio to the 0.3 power instead of the 0.4 — a lift that saturates sooner — and the best drive at 70 m/s spins at 1,629 rpm and leaves at 13.4°; let it grow as the 0.5 power and the best is 2,911 rpm at 19.1°. Halve the drag’s slope with spin, to 0.10, and the best spin rises to 3,928 rpm at 10.7°; raise it to 0.25 and the best falls to 1,585 rpm at 19.9°. The best spin is a property of the ball as much as of the swing, and it moves by a factor of two and a half across curvatures that are within the range different dimple patterns produce.

That is the aerodynamic content of the distinction ball makers draw between low-spin and high-spin balls. A ball whose lift saturates early, or whose drag rises steeply with spin, wants less spin from the club, and a player who generates a lot of spin gains carry by choosing it; a ball whose lift keeps growing rewards spin. The figure for “the” best spin of a 70 m/s drive is therefore a figure for one ball, and the 2,273 rpm here is Smits and Smith’s ball’s.

Where the remaining degrees go #

The best spin lands inside the launch monitors’ range of roughly 2,000 to 2,600 rpm for this ball speed; the best angle at that spin, 15° to 17°, is a few degrees above their eleven to fourteen. Three things this model leaves out push in that direction. The first is roll: monitors are tuned for total distance, and a shallower landing rolls further, which rewards moving down the ridge towards lower launch. The second is that a club does not deliver spin and launch independently — lofting the face raises both — so the achievable pairs are a line across the ridge, not the whole plane. The third is the coefficients themselves: Smits and Smith’s forms are one fit to one set of balls, and balls differ, by dimple pattern and by the speed at which their drag crisis completes, by more than the curvatures that decided the answer here.

Other games with the same structure #

The argument is not about golf in particular. Any ball launched for distance with backspin — a baseball driven to the outfield, a football punted, a volleyball served deep — has a lift that grows with the spin ratio and saturates, and a drag that rises with spin, and so has a best spin set by the same two curvatures and a ridge of nearly equal launches along which angle trades for spin. What differs is where on the coefficients’ curves each game sits. A baseball leaving the bat at 45 metres a second with 2,000 rpm of backspin has a spin ratio twice a drive’s, deep in the region where lift has nearly stopped growing, and a batter’s best launch, in the high twenties rather than the teens, sits where the drag and that saturated lift together put it; the terminal speed sets the best launch angle for the drag, and the spin’s curvatures move it from there. A ball whose spin ratio starts high has less to gain from spin and more to pay for it, and the best launch climbs back towards the angle drag alone would choose.

The convention: carry, rpm, ball speed #

Carry is the horizontal distance to first landing on level ground, in still sea-level air of density 1.225 kg/m³. Spin is backspin about a horizontal axis, held constant over the flight, in revolutions per minute; the spin ratio is the surface speed over the ball speed, $\omega R/V$. Ball speed is the speed leaving the club, not the club-head speed. The ball is the rule-book 45.9 grams and 42.7 millimetres. Best means greatest carry, found by golden-section search over the angle at each spin and then over the spin.

What the flight leaves out #

Spin decay, which the ball that never forgets its spin found to be a few per cent over a drive and which would lower the late spin ratio slightly. The drag crisis’s dependence on speed, which moves the drag’s level as the ball slows. Wind, which a high spin ratio makes the ball more sensitive to. Side spin and the axis tilt that a real strike adds, the subject of a ball that swings without spinning in another game. Roll, already named. And the fit’s range: the measured forms were fitted over spin ratios up to a few tenths, and the best drives here stay inside it, while the runaway flights of the fixed pair leave it long before they reach their absurd carries.

The club’s coupling of the two. A club does not set the launch angle and the spin independently: both come from the same loft and the same angle of attack at impact, and a player who raises one with a given club usually raises the other. The ridge of best pairs is therefore reachable only along the line a club and a swing allow, and fitting a driver’s loft is the business of placing that line across the ridge where it is highest, not of reaching any point on it a player chooses.

Where the curves come from #

Bearman and Harvey measured lift and drag on spinning golf balls in a wind tunnel in 1976, and Smits and Smith’s measurements of 1994 gave the fitted forms used here; the launch monitors that recommend launch conditions infer them from radar or camera tracking of real drives. The finding that an optimum spin needs both curvatures is the model’s, stated as such: with the fixed coefficients the best-spin question had no answer, and the earlier calculation sensibly did not ask it.

Still open: the drive that rolls #

The carry is half the distance on a firm fairway. The next calculation adds a landing model — the ball’s velocity and spin at impact, a bounce that reverses part of the normal velocity and lets friction act on the spin, and a roll decelerated by the turf — and asks where along the ridge of equal carries the total distance is greatest: whether roll is enough to move the best launch from 17° to the launch monitors’ 13°, and whether the best spin falls with it, as the shallower landing of a lower, spinnier drive would suggest, or rises, as the backspin’s braking on landing would.