The ball that never forgets its spin

The drag that falls as it speeds up integrates a ball’s flight through the drag crisis, and a ball that swings without spinning produces a sideways force from an asymmetric seam with no rotation at all. Neither of them carries the ball’s spin as a variable, because neither needs to.

This one does, and the reason is a claim that is made constantly and is almost exactly backwards.

The claim #

A ball curves late. A golf ball climbs and then drops sharply; a cricket ball drifts and then dips; a football bends most in the last third of its flight. The standard explanation is that the spin is dying away, and that the ball is “running out of pace” as the two decay together.

Both halves are worth testing and they need different tests. The second half is easy and correct: the speed does fall a lot. The first half is the one this essay is about.

Spin decays over a distance #

The torque that slows a spinning sphere comes from the shear on its surface, and that shear is proportional to how fast the surface is moving through the air — which is to say to the flight speed times the rotation rate, not to the rotation rate alone. Writing the spin-down torque that way and dividing by the moment of inertia gives a decay rate proportional to the flight speed.

The consequence is the pleasant one. Divide through by the speed and the equation is no longer about time at all:

The spin decays exponentially in arc length, with a constant equal to twice the ball’s mass over five times a spin-down coefficient, the air density and the square of the radius. It contains the ball’s mass, its radius, the air density and one coefficient — and it contains neither the launch speed nor the launch spin nor the time of flight.

The integrated trajectory confirms it. Running the coupled flight — drag, a saturating Magnus lift, and the spin equation — and plotting the spin against distance flown puts every point on the exponential to within five parts in ten million, which is the integrator’s error. That is not a fit; the exponential was derived and the integration checked against it.

Why the torque carries the speed #

The step that produces the whole result is the form of the spin-down torque, and it is worth defending because a different form would give a different answer.

A ball at these Reynolds numbers has a thin boundary layer over most of its surface. The wall shear in such a layer goes as the density times the velocity scale times the velocity difference across it, divided by nothing dimensional — in other words the skin-friction coefficient is roughly constant and the shear is quadratic in the local relative velocity. For a spinning ball flying forward, that relative velocity is dominated by the flight speed, and the asymmetry that produces a net torque is the surface speed. So the torque carries one factor of the flight speed and one of the rotation rate rather than two of either.

Two rival forms are worth naming. A torque quadratic in spin alone — which is what a ball spinning in still air experiences — gives a decay that is algebraic rather than exponential, and gives a golf ball a half-spin time of hundreds of seconds, which is far longer than any measurement. A torque linear in spin with a constant rate gives an exponential in time, which is how spin decay is usually quoted, and it is the same thing as this one at fixed speed.

The distinction only matters because the speed changes by a factor of two down the flight, and it is exactly the situation this collection keeps meeting: two models that agree at a fixed operating point and separate as soon as the operating point moves. That is the same trap counting what matters is about — a group that collapses two variables is only a collapse if the right two were chosen.

And the distance is enormous #

The coefficient is fixed once, against a measurement: golf balls have been observed to lose spin at about five per cent a second at seventy metres a second, and that pins C at 0.027. Everything else is the ball’s own mass and radius.

For a golf ball the memory length comes out at 1,202 metres. For a cricket ball, 1,435. For a baseball, 1,290. For a football, 424. For a table-tennis ball, 80. For a beach ball, 30.

Now compare each with the length of a shot. A drive of 230 metres is a fifth of the golf ball’s memory length, and it keeps 83 per cent of its spin. A cricket delivery is 18 metres against 1,435, and keeps 99 per cent. A free kick keeps 93 per cent. Table tennis keeps 96 per cent, not because its ball is good at holding spin — it is the second worst on the list — but because a shot is three metres long.

Only the beach ball meaningfully forgets, and it loses a third over twelve metres.

The full flight integration agrees: the golf drive covers 206 metres of arc and arrives with 84 per cent of its launch spin.

So the ratio goes up, not down #

If the spin barely falls and the speed falls a lot, then the thing that actually sets the aerodynamic force does the opposite of what the commentary says.

The Magnus lift coefficient is a function of the spin ratio — the surface speed of the spinning ball divided by its flight speed. Along the drive the spin falls by a sixth and the speed falls from 70 metres a second to 30, so the ratio rises from 0.096 to 0.189.

The lift coefficient nearly doubles over the flight. The lift force itself does not, because it also carries the speed squared and that has fallen by a factor of five — but the curvature of the path, which is what a spectator sees, is the lift over the mass divided by the speed squared, and the speed squared cancels. The curvature is very nearly proportional to the lift coefficient.

So the ball genuinely does turn harder at the end. The mechanism is the one nobody names: it is turning harder because it is slower, and the spin that produces the turn has hardly changed.

What the memory is worth #

The other side of the same measurement is what happens if the spin decay is left out entirely.

Integrate the same drive twice: once with the spin decaying and once with it frozen at the launch value, which is what a calculation with no history in it does. The carries differ by 0.94 per cent — about two metres in two hundred — and the apex by two per cent.

That is worth stating plainly because it cuts against the essay’s own subject. The spin’s memory is real, it is exactly exponential, and on a golf shot it is worth two metres. A model that freezes the spin is wrong by less than the shot-to-shot variation of the golfer.

The honest conclusion is not that the memory matters but that it has been priced, and that the number it comes to is small for a reason worth knowing: the memory length is far longer than the shot. That is the same finding as the previous section from the other side, and it is what makes the standard explanation of late swerve wrong rather than merely imprecise.

What the solver computed, and how it was checked #

A two-dimensional flight with drag, a saturating Magnus lift and a spin equation, integrated at fifty-microsecond steps.

Against a closed form. The spin along the flight must lie on the exponential in arc length, and it does to five parts in ten million. This is the check that matters, because it is the essay’s central claim and it is checkable exactly.

Against the morphology. Each ball’s memory length is recomputed from mass and radius alone and compared with the value used, which catches a transcription error in a table of six.

Against the shot. Every ball’s memory length is required to exceed its own shot length, which is the statement the whole essay rests on and would be worth knowing about if it failed for some ball.

And against the sign. The spin ratio is required to rise down the flight and the speed to fall. If either failed, the model would be producing the commentary’s picture rather than contradicting it, and the contradiction is the result.

Where the memory length comes from #

The group in the memory length is worth reading, because it is one this collection has met before under another name.

Mass over air density times radius squared is a length — it is the ballistic length, the distance over which drag substantially changes a projectile’s momentum, up to a coefficient. That the spin memory is the same group is not a coincidence: both are the ratio of the ball’s inertia to what the air can do to it per unit distance travelled.

So a ball that decelerates slowly also despins slowly, and the ranking of the six balls by memory length is nearly the ranking by how well they hold their speed. A cricket ball is dense and small and does both well; a beach ball is neither and does both badly.

The same group with a time in place of a length is the particle relaxation time of a particle is a low-pass filter, which is the Stokes number’s numerator. A tracer particle is a ball whose memory length is microns, and a beach ball is a tracer particle scaled up until its failure is visible.

And every memory number is one time over another is the general form. Here the natural version is a ratio of two lengths rather than two times — the shot over the memory length — which is the same statement in a problem where the natural clock is distance.

What actually makes a ball dip #

Having removed the spin’s decay from the explanation, something has to replace it, and there are three real mechanisms.

The speed falling. Covered above, and it is the largest for a topspun or backspun ball. The curvature goes as the lift coefficient, which goes as the spin ratio, which goes as one over the speed.

Gravity taking over. A backspun golf ball is being partly held up by its lift; as the speed falls the lift falls as the square while the weight does not, so the trajectory’s radius of curvature collapses at the end. Most of the visible “drop” of a drive is this rather than anything aerodynamic.

And the drag crisis, in reverse. A ball that launched above its critical Reynolds number and decelerates through it gains drag coefficient as it slows, which is the subject of the drag that falls as it speeds up. The effect is on the deceleration rather than the swerve, and it makes the speed fall faster at the end, which feeds the first mechanism.

None of the three is the spin dying, and two of the three are consequences of the speed dying, which is why the commentary is describing something real with the wrong cause attached.

The same correction applies to the seam mechanism. A ball that swings without spinning records that late swing needs no late anything — the sideways force is there the whole time and the accumulated displacement is quadratic in the distance flown, so it becomes visible late whatever the force is doing. Both essays end in the same place: a curve that looks late is what a constant force looks like when it is integrated twice, and no additional mechanism is needed to produce it.

The one case where the spin does die #

The beach ball is on the list to make the point that the argument is about numbers rather than about principle.

At 30 metres of memory length, a ball thrown twelve metres arrives with two thirds of its spin, and one that stays in the air for a long lofted flight arrives with much less. The same is true of a balloon, a badminton shuttle after its turnover, and any object whose mass is small compared with the air it displaces.

The dividing line is the memory length against the shot, and it is one number. Anything with a memory length of hundreds of metres keeps its spin over any shot a human can play, and everything on the list except the beach ball is in that class.

What it would take to see this on a real shot #

The claim is falsifiable with equipment that exists, and it is worth saying what the measurement would have to do, because the reason it is not routine is instructive.

Tracking spin, not inferring it. Launch monitors report spin at launch and most infer the rest from a model. A radar that resolves the spin signature through the whole flight measures it directly, and that is what would settle the decay constant for balls other than golf.

Separating the two effects. The prediction is that the lift coefficient rises down the flight while the spin falls slightly. A trajectory alone cannot separate them, because the observable is the product; what separates them is measuring the spin independently and reading the lift coefficient off the curvature.

And doing it at two launch speeds. The exponential in arc length says two shots of the same length lose the same fraction of spin regardless of how fast they were struck or how long they were in the air. An exponential in time says otherwise. That is a clean discriminating experiment and it needs one radar and an afternoon.

It has not been done for most of these balls because the answer is not commercially interesting: the correction it would supply is the two metres computed above. It remains the difference between a correct account of a late-dipping ball and a popular one, which is what a threshold that is also a duration is about in its own subject — a quantity that is quoted as a constant because nobody has had a reason to vary the thing it depends on.

What the picture cannot show #

The spin here is a scalar. A real ball’s spin axis moves — a cricket ball’s seam precesses, a football struck off-centre wobbles, and a knuckleball’s whole point is that its axis barely rotates at all. The memory length says how fast the magnitude decays and says nothing about the axis, whose behaviour is governed by the ball’s inertia tensor and by aerodynamic moments this model does not have.

The lift model is a saturating function of spin ratio fitted to the general shape of dimpled-sphere data. Its value at a given ratio is uncertain by tens of per cent, and it is used here only to produce a trajectory of about the right shape. The essay’s claims do not depend on it: the spin decay is independent of the lift model, and the ratio’s rise depends only on the speed falling.

And the trajectories are two-dimensional, so nothing here is about a ball curving sideways while also dropping, which is what most of the interesting shots do.

Where the model stops #

One spin-down coefficient for six balls. It is calibrated against golf and applied to a cricket ball and a beach ball, whose surfaces are nothing like a dimpled one. The memory lengths for the others are therefore accurate to whatever that coefficient is accurate to, which is perhaps a factor of two — and a factor of two does not change any conclusion here, because the shortest memory length among the five sporting balls is fourteen times the longest shot.

No wind. A crosswind changes the air-relative velocity and therefore the spin ratio directly, and on an outdoor shot it is usually a larger effect than anything computed here.

No bounce. Everything ends at first contact with the ground.

And no seam. The sideways force from an asymmetric seam is a separate mechanism with a separate essay, and a real cricket ball has both at once — a Magnus force from its spin and a seam force from its orientation — which can act in the same direction or in opposition depending on how it was released.

Who found it, and when #

Newton described the swerve of a tennis ball in 1672 and Magnus measured it on a rotating cylinder in 1852. The spin-decay measurements on golf balls are much more recent: Smits and Smith’s wind-tunnel work in the early 1990s produced the decay rate used to fix the coefficient here, and radar tracking of real drives since has confirmed that a driven ball retains most of its spin.

That the decay is naturally exponential in distance rather than in time is implicit in the torque model and is not usually said out loud. It is worth saying because it is what makes the comparison with the shot length the right comparison, and the shot length is a number everybody already knows.

Limits recorded rather than smoothed over #

The coefficient is fitted. One number in this essay is not derived: the spin-down coefficient, fixed against a measured golf-ball decay rate. Everything else follows from it and from the balls’ masses and radii.

The shot lengths are nominal. 230 metres for a drive, 18 for a cricket delivery, three for a table-tennis rally. They are chosen to be typical rather than measured, and the conclusions are not sensitive to them because the ratios involved are factors of ten and more.

The integrated flight is one launch condition. 70 metres a second at twelve degrees with 3,000 rpm. Other launches give other carries; the exponential in arc length is independent of the launch, which is the point.

And the two-metre carry difference is model-dependent. It comes from the lift model, which is the least trustworthy part of the calculation. What is not model-dependent is that the spin barely decays, which is where the two metres comes from in the first place.