A driver reaches the best drive only by striking upward

The run moves the best drive below the ridge that carries furthest found the longest drive in all — carry and run together, on an ordinary fairway — by searching the whole plane of launch angle and backspin. For a ball leaving at 70 m/s it is 1,359 rpm at 11.6°, landing near thirty-five degrees. The essay ended by doubting its own search. A golfer does not choose a launch angle and a spin rate. The club chooses them, both at once, from the speed of the head, the loft its face presents at the moment of contact and the direction the head is travelling, and the pairs a real strike can reach are a line across the plane, not the plane.

This essay draws the line. It needs one more piece of mechanics than the free search did — the impact itself — and with it the question becomes concrete: can a player who swings the way most players swing reach the drive the free search found, and if not, which of the two things a player controls, the loft of the driver or the angle at which it meets the ball, gets closer?

The strike, in two impulses #

Contact between a driver and a ball lasts about half a millisecond, short enough to treat as two impulses. The head is a free mass of 200 grams moving at speed $v$ along a path tilted $\mathrm{AoA}$ above the horizontal, the angle of attack. Its face is tilted so that its normal points $L$ above the horizontal, the dynamic loft — which is the loft the face actually presents at impact, not the number stamped on the club. The angle between the path and the face normal is the spin loft, $s = L - \mathrm{AoA}$, and it is the quantity everything turns on.

In the face’s own frame the head’s velocity has a component $v\cos s$ along the normal and $v\sin s$ along the face. The normal impulse obeys a coefficient of restitution of 0.83, the limit the rules of golf allow a driver, and gives the ball a speed along the normal of $(1+e)\,v\cos s/(1 + m/M)$. The tangential impulse is friction, and the standard idealisation is that there is enough of it for the ball to leave the face rolling — its contact point moving with the face. A ball with the moment of inertia of a solid sphere then leaves with a speed down the face of $v\sin s/(3.5 + m/M)$ and a backspin of $2.5$ times that over its radius. Penner wrote the model this way in 2001; Cross has measured how closely real balls follow it.

The rolling assumption is less bold than it sounds. The friction the face must supply is the ratio of the two impulses, which works out to $\tan s\,(1 + m/M)/\big((1+e)(3.5 + m/M)\big)$, about $0.18\tan s$: at a twelve-degree spin loft a friction coefficient of 0.04 is enough to bring the ball to rolling before it leaves. The coefficient between a ball’s cover and a driver’s face is several times that. A driver is a low-angle impact, and at low angles friction always wins; the ways real strikes fall short of the model are elsewhere, in the compliance of the ball’s surface, and they are taken up below.

Three consequences follow at once, and the figure below shows them.

The spin depends on the spin loft alone. A strike at 12° of loft with a level path and one at 16° of loft rising at 4° give exactly the same backspin. The launch depends on both: it is the loft minus a small angle, $\arctan$ of the ratio of the tangential to the normal speed, about a fifth of the spin loft. So the launch is close to $\mathrm{AoA} + 0.82\,s$. And the ball speed falls with the spin loft as $\cos s$, slowly: a strike delivered along the face normal sends the ball fastest.

That is the whole of the constraint. Spin costs spin loft; launch costs loft. A player who wants a high launch with little spin wants a large loft and a small spin loft, which is to say a large angle of attack. Changing the driver’s loft at a fixed attack moves along a line on which launch and spin rise together.

The level strike’s line misses #

At a level strike the line in the first figure runs from no spin at no launch up through 2,800 rpm at 9.8°, and its best drive — found by searching the loft, with the same flight, landing and run as the free search — is a loft of 9.1°, launching at 7.4° with 2,129 rpm, for 241.4 metres. The free plane’s best for a ball at the same speed, 66.6 m/s, is 1,349 rpm at 12.5°. No level strike reaches it: to launch at 12.5° from a level path needs a spin loft of about fifteen degrees, which spins the ball at 3,500 rpm.

Each curve is nearly flat near its best. At a level strike a degree more or less loft costs 0.4 metres; two degrees more costs 1.4. The curves stand above one another in order of the angle of attack — 237.0 metres at four degrees down, 241.4 level, 244.5 at four up, 245.9 at eight — and no choice of loft carries a curve up to the next. Loft decides where on its line a strike lands; the angle of attack decides which line. A player looking for distance in a driver’s loft is searching along the flat direction.

The lowest curve is cut short on the left by a condition, not by physics. Below a loft of about six degrees a downward strike launches the ball at four degrees and lands it shallower than twenty-five, where the crater-and-roll model would be asked about a skidding grounder it was never calibrated for, and the search stops there. At four degrees down the best admissible drive lands at exactly that limit; the true best there is not known from this model, and its 237 metres is a bound from below.

Upward, until about nine degrees #

With the loft chosen afresh for each angle of attack, the best total rises with the attack until about nine degrees upward at a 45 m/s head speed, and falls slowly beyond. The rise is about a metre a degree from four down to four up, and half that from four to eight.

The best strike at 45 m/s is an attack of 9.1° with a loft of 13.9°: the ball leaves at 66.8 m/s, launches at 13.0° with 1,123 rpm, carries 207.8 metres and runs 38.2 for 246.0. That is the free plane’s best within a degree and two hundred revolutions, and it is reached by a strike most players never make. Measured attack angles with a driver average a degree or two downward among tour professionals — they play the driver from a tee as though it were an iron — and five or more upward among long-drive competitors.

The gain is modest in metres and large in what it says. A level strike’s best is 4.6 metres short, under two per cent. The reason is that the total is flat near its maximum in every direction, as the free search found for the pair; a constraint that cuts across a flat maximum near its top costs little — the same reason elliptic loading is an optimum that does not matter to a designer, though it decides what the best wing looks like. But the constraint decides what the best drive looks like, and it does not look like the drive a player gets by choosing a loft.

Loft or attack, for one player #

The question the free search left can be put to a single player. Take a 45 m/s head, a driver whose face presents 10.5 degrees at impact, and the strike most club players make, two degrees downward. The drive launches at 8.2 degrees with 2,920 rpm and goes 236.9 metres. There are two things to change.

Change the loft. At two degrees down the best dynamic loft is 8.2 degrees, which lowers the spin to 2,400 rpm and the launch to 6.4 degrees, and the total rises to 239.3 metres — 2.4 metres, bought by a lower, flatter drive that runs further.

Change the attack. Keep the 10.5 degrees and strike two degrees upward instead: 1,994 rpm at 9.0 degrees, 243.1 metres. Five degrees upward: 1,293 rpm at 9.5 degrees, 244.6 metres, 7.7 more than the start. With the loft then re-chosen for the new attack, 11.5 degrees, it is 245.0.

The attack is worth three times the loft for this player, and the two do not add the way a fitting suggests. At the downward strike the better loft is lower, because the strike already spins the ball too much; at the upward strike it is higher, because the strike has taken the spin away and the launch can be bought back with loft. A loft chosen for one attack is the wrong loft for another, which is why a driver fitted to a player who then learns to hit up needs refitting with more loft, not less.

Faster heads want less of it #

The best attack falls as the head speeds up: eleven degrees at 40 m/s, 9.1 at 45, 7.3 at 50. A faster head launches a faster ball, and a faster ball’s best launch is lower because its flight is longer against its terminal speed; the attack it needs to keep its spin down at that launch is lower too. The slow swinger has the most to gain from swinging up and is the least likely to be told so, because the advice to hit up with the driver came from launch monitors fitted to fast players.

Where the distance comes from is worth separating. The best drive’s carry grows by eleven metres from four degrees down to eight up, while its run moves by a few metres and not monotonically. The upward strike does not buy a longer run by landing flatter; it buys a longer carry by launching higher with less spin, which is the trade the carry’s own best spin rested on: lift tires as spin grows and drag keeps rising, so the cheap way to height is launch, not spin. A spinning ball should leave low because a ball with lift is a glider; the best drive is a glider launched as high as its spin allows, and only the attack lets a club raise one without the other. The club makes launch cheap only through the attack.

The thirty-five degrees survive #

The free search ended with the observation that the best drive in all lands near thirty-five degrees at every ball speed, and asked whether that survives a constraint on the pair. It does. The spin that sets it is carried the whole way: the ball never forgets its spin over a flight of a few hundred metres, so the backspin the strike gives is very nearly the backspin the ball lands with.

At the best attack for each head speed the drive lands at 33.3°, 34.4° and 35.5°. Along a given speed, the landing angle of the best drive rises with the attack, from the twenty-five-degree floor at four degrees down to about thirty-four at the best. That is the reading to take: the thirty-five degrees are not a property of the free search; they are where the carry’s ridge and the run’s preference meet, and the club’s line crosses that place at the best attack. A launch monitor that reports a landing angle near thirty-five with low spin is reporting a strike near its best, whatever its loft and attack.

The convention, and what the strike leaves out #

Every number is for the same flight, landing and run as the free search: a ball of 45.9 grams, Smits and Smith’s measured lift and drag coefficients — which hold only because dimples move the drag crisis below the speeds a drive flies at — a medium fairway whose crater and roll were set so that a tour-shaped drive runs about twenty metres. The head speed is the speed of the head’s centre at impact; the dynamic loft is the face’s angle at impact, which is the club’s stamped loft plus whatever the shaft’s bend and the player’s hands add, typically a few degrees. The angle of attack is the head’s path, positive upward.

The impact model leaves things out, and one of them matters to the numbers. It is a contact between two solids, and the fluid never enters it — unlike a ball meeting a wall through a liquid film, whose rebound the film decides. A ball that leaves rolling has the most spin friction can give it; measured strikes give less, because the ball’s surface is compliant along the face and recovers some of its tangential deformation. At a tour-average strike — 50.5 m/s, a degree and a third downward, about twelve degrees of dynamic loft — the rolling model gives 3,480 rpm where launch monitors average about 2,700, so measured strikes spin at about 0.77 of the rolling model’s. Run with that share, the best strike at 45 m/s moves from 9.1 to 8.6 degrees of attack and loses half a metre. The conclusion is insensitive to the share; the absolute spins in the figures are an upper bound. The model also leaves out the face’s flexing, the head’s rotation during contact, and every off-centre strike — the gear effect of a hit high on the face, which lowers spin, is part of why players are told to tee the ball high, and it would favour the upward strike further.

And the strike is a constraint the swing imposes, not a choice it costs nothing to change. An upward attack with a driver means teeing the ball forward and the head travelling past the bottom of its arc, and whether a given player can do that without losing head speed or centred contact is a question about the player.

Who measured the strike #

The rolling-ball impact model is in Penner’s papers on the physics of golf from 2001 and 2003, which used it to explain why a driver’s best loft falls as the head speed rises; Cross measured oblique impacts of balls on heavy surfaces and found where the rolling condition holds and where the ball’s tangential compliance takes over. The practical discovery that drives go further with an upward attack came from radar launch monitors in the late 2000s, whose makers published optimal-launch charts by head speed and angle of attack; the long-drive competitors had found it first, by teeing high and hitting up.

How it was checked #

The impact conserves the momentum of head and ball along the normal and along the face to rounding, meets its restitution to rounding, and leaves the ball’s contact point moving with the face to rounding — the rolling condition imposed by the tangential impulse, checked from the velocities after it. It dissipates 5.8 per cent of the head’s energy, and never creates any. A strike along the face normal launches at the loft with no spin. Strikes at 12° of loft from a level path and at 16° rising 4° give identical spins, which is the statement that only the spin loft sets the spin. The search over loft scans 160 lofts and refines the best by golden section, rejecting drives that land shallower than twenty-five degrees; the search over attack refines by golden section on top of it. And the strike refuses a still head, a face pointing below its own path, a restitution above one, a negative mass, and more spin than a rolling ball can have.

Still open: the drive that lands on a slope #

Every drive here lands on a level fairway, and the run that makes thirty-five degrees the best landing depends on the ground’s angle to the ball’s path. A fairway sloping down and away lengthens the run of a flat landing more than a steep one, and one sloping up shortens it, so the best strike should move — towards less spin and a lower launch downhill, more of both uphill. The next calculation tilts the landing surface by a few degrees either way, reruns the crater bounce and the roll on the slope with gravity along it, and asks how far the best angle of attack moves per degree of slope at each head speed, and whether a player who cannot change the attack can recover the loss by changing the loft.