The best throw is a tangency

Forty-five degrees is the most famous number in mechanics, and most of the throws anybody makes in earnest are not at forty-five degrees. A shot putter releases at about 37°. A long jumper leaves the board at about 20°. A javelin, a discus, a football kicked for distance and a ball thrown from the back of a moving vehicle all have their own angles, and each is usually explained by an argument of its own: release height, aerodynamics, the structure of the knee, the geometry of the run-up.

Those arguments are all correct, and they are all the same argument. It becomes visible once range is drawn not against the launch angle but over the space of launch velocities, where the question “what is the best throw?” turns into a question about two shapes touching.

Range as a map over velocities #

A ball launched from the ground with horizontal velocity $v_x$ and vertical velocity $v_y$ stays in the air for a time $2v_y/g$ and covers the ground at $v_x$ throughout, so it lands at

$$R = \frac{2\,v_x v_y}{g}.$$

The range is a product of the two components, and nothing else about the launch enters. Draw the plane of launch velocities — $v_x$ across, $v_y$ up — and every curve along which the range is constant is a curve along which that product is constant: a hyperbola.

That is a map in the ordinary sense. The hyperbolae are contour lines of range, climbing towards the upper right, and a thrower is a region of the map: the set of launch velocities that particular body, arm or machine can actually produce. The best throw is the highest contour the region reaches, and the highest contour a smooth region reaches is the one it just touches.

For a thrower who can produce one speed in any direction, the region is a half-circle centred on the origin. A half-circle and a family of hyperbolae are both symmetric about the diagonal, so the tangency sits on the diagonal, where $v_x = v_y$. That is 45°, and this is the whole reason for it. The famous angle is a property of the shape of the set of throws, and specifically of its being a circle centred on the point where the ball has no velocity.

The level-ground result the angle that throws furthest derives from $\sin 2\theta$ and the one drawn here are the same calculation, and they emphasise different things. The formula says that the answer is 45°. The map says why it would stop being 45° — because any change to the shape of the region, or to the shape of the contours, moves the point where they touch.

A throw from something already moving #

The simplest change to the region is to move it. A thrower standing on a platform that moves forward at a speed $u$ produces, relative to the platform, the same half-circle as before; relative to the ground every one of those launches has $u$ added to its horizontal component. The half-circle slides sideways by $u$.

The shifted circle touches the hyperbolae higher up its own arc and lower down in the ground’s angle, and both halves of that are worth saying separately. The thrower aims steeper than 45°: the horizontal speed is already supplied by the platform, free, and the arm’s effort is better spent on the vertical component, which buys time aloft during which the platform’s speed does the travelling. The ball flies flatter than 45°, because its ground velocity is the arm’s velocity plus the platform’s, and the platform’s is all horizontal.

The tangency can be written down. On a circle of unit throwing speed centred at $(u, 0)$ the range is $2(u + \cos\theta)\sin\theta$, and setting its derivative to zero gives

$$2\cos^2\theta + u\cos\theta - 1 = 0,$$

which the figure solves and also checks by searching every direction on the circle. At $u = 0$ the root is $\cos\theta = 1/\sqrt{2}$, and the 45° comes back.

The two curves part symmetrically from 45° and keep going. At twice the throwing speed the thrower should aim at 68.5° and the ball leaves at 21.5°. In the limit of a very fast platform the aim tends to straight up — every throw’s horizontal speed is overwhelmingly the platform’s, so the arm’s only useful contribution is time in the air — and the flight tends to horizontal. Thrown backwards the pattern inverts: the platform is taking horizontal speed away, the arm has to supply it, and the aim drops while the flight steepens.

There is a question about frames hiding in this that is worth pulling out, because it looks like a paradox and is not. The ground-frame trajectory of the best throw is an ordinary parabola, and an observer on the ground could ask why the thrower did not simply launch at 45° in the ground frame. The answer is that 45° in the ground frame is not a throw this thrower can make at full speed: the circle of available launches is centred on the origin of the platform’s frame, and in the ground’s frame it is off-centre. Which launches are available is a fact about the thrower, and the thrower moves. The energy that depends on the observer makes the same point about kinetic energy: the physics is the same in both frames, and the bookkeeping of what is “free” is not.

The set of throws a body can make #

A human arm does not produce a circle either, standing still. It pushes harder forward than upward — the shoulder and the trunk are built to drive horizontally, and a steep throw has to be made against the arm’s own weight — so the achievable speed falls as the launch is raised. The set of throws is a half-circle squashed downward on its upper side.

A long jumper is a sharper case. The horizontal speed at take-off comes almost entirely from the run-up, nine or ten metres a second for an elite jumper; the vertical speed is what one leg can add in the last contact with the board, about three metres a second. Those are two separate budgets, and the set of take-offs available is, to a first approximation, a rectangle.

Neither answer is a failure of the 45° rule, and neither needs a separate theory. The squashed curve touches a hyperbola lower down because it is flatter than a circle where the circle would have touched; the tangency condition is $v'(\theta)\sin 2\theta + v(\theta)\cos 2\theta = 0$, and with the speed falling at 30 per cent per radian its root is 34.9°. The rectangle touches at its corner because a corner is where a region with no smooth boundary reaches furthest along any direction with positive components, and the corner is at $\arctan(3.2/9.5)$.

The measured angles sit close to those numbers for the reasons drawn. Shot putters release at about 37°, below the 42° that release height alone would give them. Long jumpers take off at about twenty degrees, and the attempts to jump higher by taking off steeply fail for the reason the rectangle shows: extra vertical speed has to be paid for with a slower last stride, which takes the corner inward along both axes at once. A best angle below 45° is not a compromise with an ideal; it is the ideal for the set of throws that body can make.

Drag bends the map instead #

There is a second kind of change the map can undergo, and it is the one the angle that drag moves is about. Air does not change what launches a thrower can make. It changes where they land, which is to say it changes the contours.

With drag the range is no longer a product and the contours are no longer hyperbolae; they have to be computed, direction by direction, by integrating trajectories until the one that lands at the required range is found. What comes out is a family of curves that bend away from the origin — more speed is needed for the same range — and bend most at steep angles, because a steep launch spends longest in the air and pays drag the longest. The circle of throws, untouched, now meets its highest contour below the diagonal, at 38.1°.

That number was found in the drag essay by searching the range directly, and it is found again here as a tangency on a computed map, which is a check that the two descriptions are the same statement. It is also a sharper explanation than “drag steals range”, which cannot be the reason: drag steals range from every angle, and a uniform loss moves nothing. What moves the answer is that the loss is not uniform across the map.

The two corrections now have one shape. A body that pushes harder forwards squashes the region; drag bends the contours; both move the tangency below 45°, and a real throw has both. The map separates what belongs to the thrower from what belongs to the air, which a single formula for the best angle mixes together.

A ball thrown from a train #

The platform result is easiest to believe with numbers in it. Take a throw of twenty metres a second, which on still ground goes 40.8 metres at best, at 45°. Now make the throw from the open door of a train moving forward at twenty metres a second — a platform speed of one throwing speed.

The map says to aim at 60° above the horizontal, relative to the carriage. Over the ground the ball then leaves at 30°, with thirty metres a second of horizontal speed and seventeen of vertical, stays up for 3.5 seconds, and lands 106 metres from the point where it left the hand. That is the best any throw of that speed from that train can do, and it is two and a half times the standing throw.

It is also, measured in the carriage, a poor throw. In 3.5 seconds the train has gone seventy metres, so relative to the train the ball has landed 35 metres ahead — exactly what a 60° throw reaches standing still, and less than the 40.8 metres a 45° throw would have given a passenger playing catch along the carriage. The throw is best over the ground and mediocre in the train, and nothing about it changed between the two descriptions except which frame the landing is measured in.

That is the frame question in its sharpest form. The constraint belongs to the thrower’s frame and the objective belongs to the ground’s, and the optimum depends on both: the circle of throws is centred where the arm is at rest, and the contours of range are drawn where the ball comes down. A catapult bolted to the carriage floor, whose launch energy — and so whose launch speed, in any direction — is fixed in the carriage, would face exactly the same map.

The shift of the circle is itself an approximation that a moving platform eventually exposes. Velocities add by simple vector addition only at speeds far below light; at higher speeds speeds refuse to add, and the set of velocities a platform-borne launcher can produce, seen from the ground, is no longer a shifted circle but a shifted ellipse, squashed along the direction of motion. The space that speeds live in is the geometry in which that ellipse is a circle again. No ball is thrown at those speeds, and the point is only that the shape of the set of available launches is a fact about frames, which is why moving the thrower moves the answer.

Why a tangency #

That the best point of a region is where it touches a contour is not a special fact about projectiles. It is what it means to maximise one quantity subject to a constraint, and the condition that the two curves touch is the condition that their normals are parallel — the method of Lagrange multipliers, stated as a picture. The multiplier is the rate at which the best range grows as the region is enlarged, a price in metres of range per metre per second of extra throwing speed.

The same construction runs through the rest of physics under other names, and the most surprising of them is temperature. A system in contact with a reservoir settles where its entropy is highest among the states with the energy it is allowed; drawn over the space of states, that is a tangency of a surface of constant entropy with a surface of constant energy, and the multiplier that makes the two normals parallel is $1/T$. What a system actually minimises is that statement turned into a free energy. The 45° answer and the definition of temperature are the same geometry: an optimum on a constraint, where the price of relaxing the constraint is the multiplier.

The flatness of the optimum comes from the same place. At a tangency the objective is stationary along the boundary of the region, so moving a little along the boundary costs only in second order, which is why the angle that throws furthest found that ten degrees off 45° loses almost nothing, and why every minimum is a parabola near the bottom. A thrower does not need to hit the tangency exactly; they need to be near it, and the region’s shape decides how near is near enough.

Where the map stops being right #

The launch and landing are at one height. The range is $2v_xv_y/g$ only for a ball that comes down where it left. Released above the landing ground, as a shot put is, the time of flight gains a term, the contours are no longer hyperbolae, and they tilt so as to favour flatter launches — which is the release-height correction one curve answers every slope computes, arriving here as a change of map. Both it and the body’s squashed region push the shot put below 45°, and the measured 37° is their combined result.

The body’s region is a model. A speed falling at thirty per cent per radian is a parameter chosen to show the effect, not a measurement of any athlete, and the long jumper’s rectangle ignores the most important trade in the event — that planting the take-off foot to gain vertical speed costs horizontal speed. A measured region for a real thrower is a closed curve with its own shape, found by filming many throws at many angles, and the tangency on it is the answer for that person.

The platform moves steadily. A thrower on an accelerating or turning platform has a region that changes during the throw, and in a rotating frame there are forces on the ball after it leaves the hand that this map does not contain.

And range is not always what is wanted. A throw to a moving target, a throw that must clear a wall, a kick that must stay in the air long enough for players to arrive under it each maximise something else, and each draws a different family of contours on the same plane.

What the map cannot show #

The velocity plane has no time in it. Two launches on the same hyperbola land at the same place and are nothing alike: one arrives in a fraction of a second at a shallow angle, the other climbs for twice as long and falls steeply. Everything about when and how a throw arrives has been projected away, and the map is only as good as the decision to care about range alone.

Nor does it show variability, which is what a real thrower actually has. A region drawn as a sharp curve is the set of best efforts, and every real throw lands somewhere inside it, scattered. Near the tangency the scatter in angle costs little and the scatter in speed costs a great deal, and a thrower whose speed is erratic might sensibly aim a little away from the tangency to trade one for the other — a decision about the distribution of throws that no drawing of a single region can make.

Still open: the best throw when range is not the aim #

The contours drawn here are range, and the tangency rule works for any quantity that can be drawn as contours on the velocity plane. Height at a given distance, time aloft, arrival speed and the probability of passing through a window of a given size are all such quantities, each has a family of curves on the same plane, and each gives its own best throw from the same region of available launches.

The case with the most in it is the one where the quantity to maximise is a probability: a throw at a target, from a body whose launches scatter in both speed and angle, where what is wanted is not the farthest throw but the one most likely to arrive. There the region is blurred, the contours are of a probability rather than of a distance, and the answer depends on how the scatter in speed compares with the scatter in angle — which is a question about a particular thrower as much as about mechanics.

The habit worth carrying away is the separation the map forces. When an optimum moves, ask whether the constraint changed or the objective did. A body pushing harder forwards and air resisting the flight both lower the best angle, and they do it in completely different ways — one reshapes what can be done and the other reshapes what it is worth — and only a picture that draws both separately shows which is which.