A spinning ball should leave low

The terminal speed sets the best launch angle found that a ball with drag and no lift goes furthest at an angle below 45°, and that the angle depends only on $\beta$, the square of its launch speed in terminal speeds. At the speeds games are played at, $\beta$ is one to three and the best angle is 37° to 41°. The essay ended on the assumption it had made to get there: that the ball does not spin.

Almost every struck ball does. A golf ball leaves a driver with two to three thousand revolutions a minute of backspin, a baseball off a bat with one or two thousand, a tennis ball sliced or a football chipped with comparable rates. Lift with no wing at all is the account of what that spin does to the air: it puts circulation round the ball, and circulation in a stream is lift. The ball that never forgets its spin found that the spin barely decays over a flight. So the lift is there from launch to landing, and the question here is what it does to the best angle. The answer is that it does far more than the drag did.

A ball is a glider #

Write the lift as a fraction of the drag: $C_L = \ell\,C_D$. The lift is then $\ell$ times the drag force, directed across the path rather than against it, and in the drag-length units of the earlier essay the equation of motion becomes

$$\frac{d\mathbf{v}}{dt} = -\hat{\mathbf{y}} - |\mathbf{v}|\,\mathbf{v} + \ell\,|\mathbf{v}|\,\hat{\mathbf{z}}\times\mathbf{v}.$$

Two numbers are left: $\beta$, as before, and $\ell$, the ball’s lift-to-drag ratio. That is the number by which an aircraft’s glide is measured, and the cheapest way to stay up is about how a wing makes it large; a ball makes it small — a golf ball’s is about 0.6 at launch, a baseball’s somewhat less — but small is not nothing. A golf ball with $\ell = 0.6$ at $\beta = 2.39$ starts its flight with a lift of $\ell\beta = 1.4$ times its own weight. It would climb if launched level.

Holding $\ell$ fixed along the flight is a model, and the essay says where it is wrong below. It is the model that makes the picture two-dimensional, and the right first model for the same reason a constant drag coefficient was: it isolates the effect from the details of how a spinning ball’s lift coefficient depends on its spin ratio.

Lift moves the angle faster than drag does #

The bottom curve of the figure is the earlier essay’s: drag only, falling slowly from 45°. Every curve above it is lower, and they are much lower. At $\beta = 10$ the drag alone puts the best angle at 32.6°. A lift-to-drag ratio of 0.4 — less than half of any glider’s worst — takes it to 15.9°. At 0.8 the best launch is level.

The pattern is that lift does to the angle in one tenth of its range what drag needed three decades of $\beta$ to do. The reason is that the two act on the angle by different routes. Drag lowers the angle indirectly: it makes the descent steep, which devalues height. Lift lowers it directly: it supplies height. Every degree of launch angle is height bought by pointing the speed upwards, which costs horizontal speed and therefore range; lift supplies height without costing any speed at all — it is perpendicular to the path and does no work — and it is strongest at launch, where the speed is. A thrower who gives the ball an angle the lift would have given it for free has paid twice.

A fixed slice per tenth #

Read the other way, at a fixed $\beta$, the angle is nearly a straight line in $\ell$ until it reaches level. Each tenth of lift-to-drag ratio takes 1.3° off at $\beta = 1$, 2.2° at $\beta = 2.39$ and 4.1° at $\beta = 10$. The slope grows with $\beta$ for the reason given above: lift goes as the square of the speed, so a ball launched at many terminal speeds has a lift at launch many times its weight, and the angle it can do without grows in proportion.

The straightness is not exact — the curves bend slightly towards the end — but it is close enough to be a rule. A golf ball whose spin is changed by a club, a ball or a strike, enough to change its lift-to-drag ratio by 0.1, should be launched about two degrees lower or higher to stay at its best; a fitter adjusting the loft of a driver by a degree is trading exactly that.

Where the best launch becomes level #

Each curve in the first figure reaches zero and stays there. Beyond that point any upward angle at launch loses range, because the lift climbs the ball more cheaply than an angle would, and the best flight leaves level. The figure is the boundary. At $\beta = 1$ it takes a lift-to-drag ratio of 2.33 — more than any ball has — so a ball launched at its terminal speed always wants some angle. At $\beta = 10$ the boundary is at 0.746, and at $\beta = 100$ at 0.34.

A golf drive, with $\ell$ a little over a half and $\beta = 2.39$, is well below the boundary of 1.44 at its $\beta$: its best launch is low, not level. A table-tennis topspin loop is a different case altogether — topspin is negative lift, and it moves the angle the other way. A badminton shuttle has no lift. The balls that could cross the line are light, fast and heavily spun ones, which is why a hard-hit backspin shot in table tennis can be struck almost flat and still rise.

The boundary also says what “level” means here. It is not that the ball flies level: it climbs, sometimes steeply, because its lift at launch is many times its weight. It is that no angle given to it at launch would improve on the climb the lift supplies.

A drive launched at the wrong angle #

The same thing in a real ball’s units. Here the lift is the classical Magnus form instead of a fixed ratio: $C_L = 1.6\,S$ with the spin ratio $S = \omega R/V$, so the lift force grows as $\omega V$ rather than as $V^2$, and the spin rate is held at 3,000 revolutions a minute throughout. The coefficient 1.6 is a round figure inside the range wind-tunnel measurements on golf balls give at low spin ratio; it is stated, not measured here. The drag coefficient is 0.25, as before.

Unspun, the ball’s best angle is 38.1° and it carries 202 metres. Give it 3,000 revolutions a minute and launch it at the same 38.1°, and it balloons: it climbs to 98 metres, hangs, and comes down 213 metres away — eleven metres further than the unspun ball, for the price of a flight nearly twice as high. Launch the same spinning ball at its own best angle, 17.1°, and it carries 262 metres on a flight half as high. The spin was worth sixty metres; launching it at the unspun ball’s angle threw five-sixths of that away.

A golfer who has watched a high-spinning drive climb, stall and drop has seen the middle curve. The fix that coaching literature gives — lower the launch or reduce the spin — is the arithmetic of this figure, and the second half of it is the part this model cannot reach, below.

What the lift buys, and what the wrong angle gives back #

The figure separates the two halves of the drive’s story at a fixed $\beta$. The dark line is what lift is worth when the angle is chosen for it: twelve per cent more range at $\ell = 0.6$, twenty-one per cent at $\ell = 1.2$. The warm line is what is lost by keeping the unspun angle: eight per cent of the best at $\ell = 0.6$, thirty at $\ell = 1.2$.

Multiplied together they say whether spin launched at the wrong angle is worth having at all. At $\ell = 0.6$ it is, just: the spinning ball at the unspun angle still carries 2.6 per cent further than the unspun ball at its best. At $\ell = 0.74$ the two are equal, and past that the spin launched at the old angle is a net loss — at $\ell = 1$ it carries seven per cent less than no spin at all. This is the quantitative version of the coaching rule, and it says that the rule matters more the more the ball spins.

It is also a warning about how a flat optimum can mislead. The earlier essay found that missing the drag-only optimum by five degrees cost about one per cent, so the angle hardly mattered. Missing the lifting optimum by twenty degrees — which is what keeping the unspun angle does — is far outside the flat top, and the cost is no longer small.

A fly ball, and the descent the lift flattens #

A baseball shows the same structure at a lower $\beta$. Hit at 45 metres a second with a drag coefficient of 0.35 it has a drag length of 159 metres and $\beta = 1.30$; unspun, its best angle is 40.2° and it carries 110.5 metres. Backspin off the bat gives it a lift-to-drag ratio of a few tenths. At $\ell = 0.2$ its best angle is 37.0° and it carries 114.2 metres; at 0.4, 33.7° and 118.0. Four tenths of lift-to-drag are worth seven and a half metres and six and a half degrees, which is the difference between a warning-track out and a home run in most parks. The direction matches what tracking systems report for the longest hits, which leave the bat in the high twenties and low thirties rather than near forty; the size of the remaining gap is the borrowed drag coefficient, whose true value for a baseball scatters between 0.3 and 0.5 from ball to ball.

The lift changes the other end of the flight too. Without lift a ball’s best flight lands much steeper than it leaves, which was the earlier essay’s account of why the angle falls. Lift on the descent points up and forward — it is perpendicular to a path that is now heading down — so it holds the ball up and flattens the fall. The baseball’s landing angle falls from 54.5° unspun to 50.6° at $\ell = 0.4$; the golf ball’s at $\beta = 2.39$ from 57.6° to 50.7° at $\ell = 0.6$ and 46.6° at $\ell = 1$. A flatter landing carries the ball further after it touches down on a fairway, which is the roll this calculation does not count, and it is part of why a low, spinning drive is worth more on a course than its carry says.

So lift works on the best angle from both ends, and the two ends pull in opposite directions. At launch it supplies the height an angle would have bought, which favours a lower launch. On the descent it makes height worth more, because a flatter fall turns height into distance; on its own that would favour a higher launch, undoing part of what drag did. The figures say which wins: the angle falls with $\ell$ at every $\beta$, and fast, so the free height at launch outweighs the better use of height at the end. The flatter landing is a bonus the low launch collects, not the reason for it.

Spin rate as the knob #

In the Magnus model the spin rate is the control, and the figure sweeps it. With no spin the best angle is 38.1° and the carry 202 metres. At 2,000 revolutions a minute, 25.4° and 241 metres; at 3,000, 17.1° and 262 metres. The angle falls almost linearly with spin and the carry rises almost linearly, which is the lift-to-drag picture again, since the lift-to-drag ratio at launch is proportional to the spin rate.

Neither curve turns over in this model, and that is its most important failure. On a real golf ball the drag coefficient rises with spin — a spinning ball’s wake is wider and its separation asymmetric — so past some spin the extra drag costs more range than the extra lift buys, and the carry has a maximum. Launch monitors put the best driver spin for a fast swing near 2,000 to 2,500 revolutions a minute, at launch angles of eleven to fourteen degrees. Both numbers are lower than this model’s, and both are in the direction a rising drag would push them: a ball that pays for its spin in drag wants less spin, and a ball with less lift wants — by the rule above — a higher angle than a lift-only model’s, which the spin-drag model’s lower spin then brings back down. The model gets the direction and the size of the effect, and not the optimum.

How the flights were checked #

The flights are the earlier essay’s integrator with a lift term, and the checks are aimed at that term. Lift must do no work, so the energy the ball loses must equal the work of the drag alone: it does to one part in a thousand million at $\ell = 0.6$, and to four parts in ten thousand million for the Magnus drive at 3,000 revolutions a minute. A lift of $10^{-9}$ of the drag gives the same best angle as no lift, 38.143°, so the lifting calculation joins the drag-only one continuously. And the two lift models — a fixed ratio, and a Magnus lift growing with speed rather than its square — must give the same lift at launch when their coefficients are matched there, and they do, exactly; they part along the flight as the ball slows, which is the difference the drive figures show.

The convention: lift across the path, spin held #

Lift is perpendicular to the velocity and positive for backspin, so it points up and back on the way up and up and forward on the way down. The lift-to-drag ratio $\ell$ is held fixed along a flight in the first four figures; the Magnus figures hold the spin rate fixed instead, which is what the spin’s long memory justifies. The drag coefficient is fixed in both, the ground level, the air still, and the range is measured to the launch height. A ball landing on a fairway rolls, and the roll — longer for a lower descent, which lift gives — is a further reason the low launch wins on a golf course that this calculation does not count.

What the picture cannot show #

The lift coefficient is the borrowed number, and it is borrowed in a crude form. A real spinning ball’s lift depends on the spin ratio nonlinearly, saturating at high ratios; it depends on the surface, which is why a scuffed ball and a dimpled one fly differently; and at low spin some balls show a reversed Magnus force for a band of Reynolds numbers, the same boundary-layer asymmetry that makes a ball swing without spinning. None of that is here. What the figures establish is the structure — a ball is a glider of small lift-to-drag ratio, and its best angle falls with that ratio faster than with anything drag can do — and the size of the effect for stated coefficients. The optimum spin, the optimum launch for a particular ball and the roll after landing all need measured coefficients that this calculation does not have.

Who found it, and when #

Magnus measured the force on a spinning cylinder in 1852, and P. G. Tait argued through the 1890s that the long carry of a golf ball was made by its underspin, against the then-common view that a drive was a parabola spoiled by drag, and that a drive without spin could not reach the distances golfers achieved. Bearman and Harvey’s wind-tunnel measurements of 1976 gave the lift and drag of spinning golf balls as functions of spin ratio, which is what every later trajectory model borrows. The equivalence between a lifting ball and a glider is in the equations and is older than any of them; the observation that launch angle and lift trade at a nearly constant rate is what a modern launch monitor measures every day, and what a sailing boat’s polar shows for the same reason — two drag angles and a lift deciding a best heading.

Still open: a lift that changes along the path #

Both models here hold something fixed that a real ball does not. A spinning ball’s lift coefficient depends on its spin ratio, which rises as the ball slows; its drag coefficient rises with spin; and at the top of a high flight the spin ratio can reach values where the lift saturates. The next calculation replaces both fixed coefficients with measured functions of the spin ratio — lift and drag together, as a wind tunnel reports them — and asks for the optimum pair of launch angle and spin rate at a given ball speed: whether it sits where launch monitors place it, and how much of the difference between this model’s 17° and the measured eleven to fourteen is the rising drag and how much the saturating lift.