The throw most likely to go in

Every earlier argument about the best throw has asked for an extreme: the farthest range, the least speed, the highest reach along a slope. Those questions have sharp answers because the quantity being optimised is a property of a single throw. A basketball free throw asks something else. Nobody cares how far it goes or how cheaply it gets there, only whether it goes in — and whether a throw goes in is not a property of the throw that was intended but of the throws that actually happen, which scatter.

That changes the kind of answer available. The best aim is no longer where one curve touches another; it is where a blurred patch of throws overlaps most of a region of throws that succeed. The two shapes have different origins — one belongs to the thrower’s body, the other to the geometry of a ball, a hoop and gravity — and the best aim depends on both at once.

What makes the free throw the right example is that it already has a famous answer to the wrong question. The least-speed launch to any point bisects the angle between the vertical and the line to that point, which for a release 0.95 metres below and 4.19 metres short of the centre of the hoop is 51.4°, at 7.17 metres per second. That launch has a genuine virtue for accuracy, described below. It is still not the throw most likely to go in.

The launches that succeed #

Every free throw is a launch angle and a launch speed, so every free throw is a point on a plane. For each angle there is exactly one speed that sends the centre of the ball through the centre of the hoop, and those speeds make a curve with a minimum at the least-speed launch. Around that curve lies a band: the throws that pass cleanly through the hoop without touching the rim, found at each angle by increasing and decreasing the speed until the ball just touches.

The band is the whole answer to “which throws succeed”, and it has a shape worth reading before any thrower is placed on it. It does not exist at all below 47°. Near the bottom of the curve it is a hair thick. It thickens steadily as the launches steepen, and at the same time it tilts, because the speed needed rises more and more steeply with angle.

The two ellipses are a thrower who scatters by 5 centimetres per second in speed and by one degree in angle — values chosen to show the effect rather than measured from anybody. Aimed at the least-speed launch, the ellipse lies across a band far thinner than itself, so most of its throws miss above or below. Aimed at 58.5°, more of it lies inside the band, although the band there is steeper and the ellipse’s left and right edges fall off it. The best aim is the one that balances those two losses, and the picture already says it will be steeper than the cheapest throw.

How big the hoop is depends on how the ball arrives #

The band thickens with angle for a reason that has nothing to do with the thrower and everything to do with the target. A hoop is 45.7 centimetres across and a ball is 23.9. A ball coming straight down has its centre’s worth of clearance — 10.9 centimetres either side of the middle. A ball coming down at an angle does not: in the plane of the rim it presents a longer footprint, its radius divided by the sine of the angle of descent, and the room left for its centre shrinks accordingly.

The least-speed launch comes down at 38.5°, which leaves the centre of the ball ±3.7 centimetres of room. A launch at 60° comes down at 52° and leaves ±7.7 — more than twice as much. Below a descent of 31.5° the footprint is wider than the hoop and nothing goes cleanly through at all, which is where the band in the first figure ends.

That is the whole of the target’s contribution, and it points in one direction only: steeper is better. A steep throw arrives at a hoop that is effectively larger. It is the same reason a letter dropped into a slot goes in more easily from above than from the side, and it is a fact about horizontal openings rather than about basketball.

The two windows, and why they peak in different places #

The thrower’s contribution is a pair of errors, and they behave differently along the family of throws. The figure separates them: for each aim through the centre of the hoop, how wide a range of launch speeds still goes in with the angle held perfect, and how wide a range of angles still goes in with the speed held perfect.

The speed window does what the hoop’s geometry predicts. It starts at almost nothing where clean passes first become possible and grows steadily, to 22.6 centimetres per second at 76°, because every increase in steepness brings the ball down into a larger effective opening. A thrower whose only error is in speed should aim as steeply as can be managed.

The angle window does something else. It is widest near the least-speed launch, reaching 7.8° across, and falls away on both sides. The reason is the property that made the least-speed launch famous. At the least-speed launch the centre of the hoop lies on the envelope of every throw at that speed — the curve no throw at that speed can cross — and on an envelope, a small change of angle moves the trajectory along the envelope rather than off it. The ball’s arrival point is stationary with respect to the angle. That is the flatness of an optimum put to use: an error that enters only at second order barely registers.

The double hump near the bottom of the lower curve is the same fact seen closely. At exactly the least speed, the flat and lobbed throws through the centre are the same throw; just above it they are two distinct throws with angle windows of their own, and the window measured around one of them merges with the other’s. It is a real feature of the geometry and not a numerical accident, and it only matters to a thrower whose scatter in angle is small compared with a couple of degrees.

So the two windows are widest at different aims. One wants the least-speed launch; the other wants the steepest throw available. No single aim is best for both, and which matters more depends on the thrower.

The chance of going in #

Putting the scatter and the band together gives a number: the fraction of a thrower’s attempts, aimed at a given angle, that pass cleanly through. It is computed here in two ways that share nothing but the trajectory. One counts hits over a fine grid of both errors at once. The other takes each error of angle in turn, finds the exact window of speeds that succeed at that angle by bisection, and weights it by the scatter in speed in closed form. The curves use the second, and the best value of each was checked against the first.

For the thrower of the first figure, scattering ±0.05 metres per second and ±1°, aiming at the least-speed launch puts 37 per cent through cleanly and aiming at 58.5° puts 58 per cent through. The curve near its peak is flat — anything from about 56° to 61° does almost as well — which is the same flatness that makes 45° forgiving and is the practical reason a thrower does not need to know the number.

The other two curves show which way each error pulls. Doubling the scatter in speed, to ±0.1 metres per second, moves the best aim up to 64° and cuts the success rate to 41 per cent: with speed now the dominant error, the geometry’s preference for steep throws wins more of the argument. Doubling the scatter in angle instead, to ±2°, pulls the best aim back to 55°, towards the aim where angle errors cost least, and the rate falls to 45 per cent. In both cases the aim moves towards whichever window protects against the error the thrower actually makes.

This is the answer to the question the tangency construction left open, and it has a different character from every answer before it. A best throw for range belongs to a body’s set of possible launches and to the physics; a best aim for success also belongs to the distribution of a body’s launches, which differs from one person to the next and from one day to the next. The question “what angle should a free throw be taken at?” has no answer until the thrower’s scatter has been measured.

A board instead of a hoop #

The preference for steep throws came entirely from the hoop being horizontal. Replace it with a vertical target of similar size — a board 20 centimetres high, centred at exactly the same point in space — and ask the same thrower the same question.

For the board the chance peaks at 47.0°, where 93 per cent of throws strike it, and it stays high all the way down to the flattest launch an arm limited to 11 metres per second can make. For the hoop it peaks at 58.5°. The same scatter aimed at the same point gives best aims 11.5° apart, on opposite sides of the least-speed launch.

Both targets share the least-speed launch’s virtue — at that aim, neither minds a small error of angle, because the centre of each target sits on the envelope of throws at that speed. What separates them is the error in speed. At a board, a faster, flatter throw spends less time in the air, so a given error of speed becomes a smaller error of height by the time the ball arrives, and the board is the same size however the ball comes in. At a hoop, a steeper throw makes the opening larger, and that gain outweighs the extra time aloft. The target’s orientation reverses the direction in which the speed error pushes the aim.

A dart player throws flat and fast at a vertical board; a free-throw shooter lofts the ball at a horizontal ring. Both habits are usually put down to tradition or feel, and both are what the geometry of the target asks for.

The same trade at other targets #

The argument has three ingredients, and none of them is specific to basketball: a family of launches that reach the target’s centre, a target whose effective size depends on how a projectile arrives, and a thrower whose errors are spread in more than one direction. Wherever all three occur, the best aim sits between the launch least sensitive to one error and the arrival that enlarges the target, and moves between them according to which error dominates.

A golf putt is the version in which the target’s size depends on arrival speed rather than arrival angle. A ball rolling slowly over the edge of the hole drops in almost anywhere across it; a ball arriving fast drops in only if it is aimed near the middle, and above a certain speed it crosses the hole entirely. The effective hole therefore shrinks as the arrival speed rises, which is the mirror image of the hoop growing as the descent steepens. A putt struck too softly, meanwhile, never reaches the hole at all, however well aimed. The best putt balances a target that shrinks with speed against a scatter in speed that makes a gentle putt stop short, and it arrives with a little more speed than the least that would reach — for the same structural reason that the best free throw is a little steeper than the cheapest.

A penalty kick in football is the version with a vertical target and an opponent. The goal is a board, and a board, as the last figure showed, rewards a faster and flatter delivery than a hoop does. A football struck at twenty-five metres per second covers the eleven metres in under half a second and drops about a metre on the way, a correction made by feel, and the question reduces largely to direction. The goalkeeper adds a second scatter that the kicker is trying to exceed rather than to fit inside, which turns the problem into one about two distributions rather than one, and is the point at which physics hands it over to game theory.

The comparison also says what the free throw is not like. A throw’s errors do not grow the way errors in a chaotic system double on a schedule. Over a flight of a second, an error of launch speed becomes an error of position in proportion to itself, and nothing in a projectile’s motion amplifies it exponentially. That is precisely why the question of best aim has a stable answer at all: the sensitivity of the arrival to each error is a smooth function of the aim, computable once and for all, and the only thing that varies from one person to another is the scatter it is multiplied by.

What the calculation leaves out #

The ball goes in only cleanly. A throw that touches the rim is counted as a miss here, and in practice a good fraction of such throws drop in, particularly those that strike the back of the rim with the ball descending steeply. Counting them would widen the band on its steep side more than on its shallow side and push the best aims somewhat further up, not reverse them.

There is no spin, no drag and no backboard. Backspin makes a ball that strikes the rim lose speed and fall more often, which is a second reason steep throws succeed more than a clean-pass count suggests. Drag on a basketball at about 7 metres per second is roughly a tenth of its weight — enough, for a heavier drag, to move the best angle for range — which raises the required launch speed by a few per cent and shifts every curve without changing which way the best aim moves.

The scatters are stated, not measured. The three throwers are pairs of numbers chosen to show which way each error pulls. A real shooter’s errors are neither independent nor Gaussian — a throw released high is often released faster too — and a correlated scatter tilts the ellipse, which can move the best aim in either direction depending on the sign of the correlation.

Aim here means aiming at the centre. A thrower can also aim deliberately long or short, sending the ball through a point off the middle of the hoop so that the more likely error lands in the larger part of the opening. That adds a second variable to the optimisation, and for a thrower whose errors are lopsided it is a genuine further gain rather than a refinement.

What the curves cannot show #

Each curve is a single number per aim: the fraction of attempts that succeed. It says nothing about how those successes are distributed within the hoop, how near the misses come, or how a thrower’s scatter changes under fatigue or pressure — which is exactly where the practical interest in free throws lies. A curve with a flat top says that a range of aims is equally good on average, and cannot say that one of them degrades more gracefully when the thrower’s own spread widens.

Nor do the figures show any of this in time. A throw is drawn as the launch that produced it and the place it crossed the rim, with the flight between collapsed away. The action that knows where every path ends is the description in which the whole of that flight is a single function, and in it the envelope that protects the least-speed launch from angle errors is the fold where two throws arrive at one place.

Still open: the best aim for a scatter that learns #

Every thrower in the figures has a fixed scatter. A person practising a free throw does not: the scatter shrinks with repetition, and it probably shrinks differently in speed and in angle and differently at different aims. If practice at a steep aim reduces the speed error faster than practice at a flat one, the best aim to practise at is not the best aim for today’s scatter. Motor-control research studies how people choose among movements with different sensitivities to their own noise, and there is evidence that people drift towards movements whose outcome is least sensitive to the noise they have. Whether the free-throw angles skilled shooters settle on are the ones this calculation predicts for their measured scatter — or the ones that minimise it — is a question about people, answerable only by measuring both.

The habit worth carrying away is to ask of any “best” what it is best for. The extreme of a quantity and the most probable success are different questions, and the second one depends on how the attempts are spread — which is a fact about the one making them. A launch that is cheapest, or furthest, or least sensitive to one kind of error is a property of the physics; the launch most likely to work is a property of the physics and of the person together, and a quantity built from many small random contributions is best described by its distribution rather than by its central value.