The terminal speed sets the best launch angle

Forty-five degrees is one of the few numbers in mechanics that most people know without knowing where it comes from. It comes from a parabola. A ball launched at speed $U$ and angle $\theta$ in a vacuum stays up for $2U\sin\theta/g$ and covers $U\cos\theta$ every second, so it lands $U^2\sin 2\theta/g$ away, and $\sin 2\theta$ is largest at $\theta = 45°$. The derivation has no air in it, and neither does the answer.

Every ball anyone has thrown for distance has had air round it, and the question this essay asks is the obvious one: what does the air do to the best angle? The usual reply is that drag makes everything shorter, which is true, and that the angle is a detail, which is not. The angle moves, it moves by a definite amount, and the amount is set by one number that can be read off any ball with a stopwatch and a drop test.

A ball has a length #

The drag that falls as it speeds up spent its time on the drag coefficient, because that essay was about the narrow band in which the coefficient changes. Outside that band — and a golf ball in flight, a baseball off the bat and a football off the boot all sit outside it, on the far side of the crisis — the coefficient is close to constant, and the drag force is simply $\tfrac12\rho C_D A V^2$. Divided by the ball’s mass it is a deceleration $V^2/L$, with

$$L = \frac{2m}{\rho\,C_D\,A}.$$

$L$ is the ball’s drag length: in level flight with nothing else acting, the air takes away a factor of $e$ of the ball’s speed in every distance $L$ it covers. A shot put’s drag length is 2.2 kilometres. A golf ball’s is 209 metres, a baseball’s 159, a football’s 74, a table-tennis ball’s 7.8, and a shuttlecock’s — read from its measured terminal speed of 6.8 metres a second rather than from a coefficient nobody measures — 4.7. The ball that never forgets its spin found that spin decays over a length of the same construction, and for the same reason: the air acts on a ball in proportion to how far the ball goes through it, not to how long it takes.

One number left #

Measure every length in drag lengths, every speed in $\sqrt{gL}$, every time in $\sqrt{L/g}$, and the equation of motion of a ball with quadratic drag becomes

$$\frac{d\mathbf{v}}{dt} = -\hat{\mathbf{y}} - |\mathbf{v}|\,\mathbf{v},$$

which has no parameter in it at all. The only place the ball’s identity can still enter is the launch: a speed $U$ becomes $\sqrt{\beta}$ in these units, with

$$\beta = \frac{U^2}{gL}.$$

That number has a plainer reading. The speed $\sqrt{gL}$ is exactly the ball’s terminal speed $v_t$, the speed at which drag equals weight, because $g = v_t^2/L$ there. So $\beta = (U/v_t)^2$: the square of the launch speed measured in terminal speeds. Two balls with the same $\beta$ fly the same shape, scaled by their drag lengths, and have the same best angle. That is the whole of the dimensional argument, and counting what matters would have predicted that one group is all that is left once the drag coefficient is taken as fixed; what it cannot say is what the best angle does as that group changes. That needs the flights.

The curve every ball sits on #

The answer is the curve above. At very small $\beta$ the air does nothing and the best angle is 45°. As $\beta$ grows the angle falls, at first by about 5.7° for each unit of $\beta$ — 44.43° at $\beta = 0.1$, 40.98° at 1 — and then more and more slowly: 32.63° at 10, 25.15° at 100, 20.34° at 1000. A thousandfold increase in $\beta$ beyond the first unit buys only another twenty degrees, and there is no sign of the curve levelling at any particular value.

The balls fall where their $\beta$ puts them. A men’s shot of 7.26 kilograms put at 14 metres a second has a terminal speed of 148 metres a second, a $\beta$ of 0.009 and a best angle of 44.95°: for a shot, the air is irrelevant to the angle. A baseball hit at 45 metres a second and a football kicked at 30 sit together near $\beta = 1.3$ and 40.2°, because both are launched at a little more than their terminal speed. A golf ball struck without spin at 70 metres a second — far past the crisis that a dimpled surface brings forward — has $\beta = 2.39$ and a best angle of 38.1°. A table-tennis ball at 15 metres a second, at 1.7 terminal speeds, is at 37.4°. A shuttlecock cleared at 30 metres a second is at 4.4 terminal speeds, $\beta = 19.5$, and its best angle is 30.2°.

The surprising thing about the list is what it does not depend on. A shuttlecock weighs five grams and a football four hundred and thirty; one is a cone of feathers and the other a sphere; nothing about their construction is the same. All of that is inside $v_t$, and once the launch speed is divided by it the two are on the same curve fifteen degrees apart only because one is launched at four terminal speeds and the other at one.

Reading β off a real ball #

The number is easier to measure than either of its ingredients. A drag coefficient needs a wind tunnel and an area that is not obvious for a shuttlecock or a seamed ball; a terminal speed needs only a drop from a height long enough for the ball to stop accelerating, and the speed it arrives at. A raindrop’s terminal speed is the classic case — the drop that is not a tear found it set by a balance of the same two forces — and a shuttlecock’s 6.8 metres a second was measured in exactly that way. Square the ratio of launch speed to that speed and the best angle can be read off the figure without knowing anything else about the ball.

The first unit of $\beta$ is where the angle moves fastest, and the reason is visible in the flights. A little drag slows the ball on the way up and again on the way down, but the way down happens later, at a lower speed, so it is slowed less; the descent becomes steeper than the climb, the far half of the flight shortens more than the near half, and the range gained by launching flatter outweighs the height lost. That is a first-order effect in the drag, which is why the angle falls linearly at first, at about 5.7° per unit of $\beta$. It is the same structure as one number picking a machine: the group says which regime the object is in before any detail of its design has been looked at.

What a best flight looks like #

The shapes say why the angle falls. At $\beta = 0.01$ the best flight is the vacuum parabola, symmetric about its top. At $\beta = 1$ it already leans: it climbs further than it falls. At $\beta = 100$ the ball leaves at 25.2°, reaches its highest point 71 per cent of the way along, and comes down at 70.7°. The flight has two parts. In the first the ball is much faster than its terminal speed and the air takes its speed away over a few drag lengths; in the second it has lost nearly all of its horizontal speed and falls, close to its terminal speed, almost vertically.

That second part is the key to the angle. Whatever the launch, a ball launched at many terminal speeds ends its flight falling steeply, because that is what a ball near its terminal speed does. The height it climbs is going to be spent on a near-vertical descent that adds little range. So height is worth less, and horizontal speed at launch is worth more, the higher $\beta$ is — and the best angle trades the one for the other at a lower angle as $\beta$ grows.

The angle matters less than its shift suggests #

Before reading the angle as advice it is worth seeing how much it is worth. Range as a function of launch angle has a flat top, as every smooth maximum does, and the flatness survives the air. At $\beta = 10$ a launch five degrees below the best angle loses 0.98 per cent of the range, and five degrees above loses 0.91 per cent. A thrower who misses the optimum by five degrees has not lost a meaningful distance at any $\beta$ on the curve.

Further from the optimum the curves stop being symmetric, which the vacuum’s are exactly. At $\beta = 100$, fifteen degrees too low costs ten per cent of the range and fifteen degrees too high costs 6.7 per cent. The low flight is the expensive mistake: it reaches the ground still carrying speed it never had time to spend, while the high flight at least spends its extra time aloft drifting forward, slowly.

The second figure puts a price on both effects across the whole curve. The warm line is what the air takes: at $\beta = 1$ the best flight reaches 59 per cent of the vacuum range, at $\beta = 10$ seventeen per cent, at $\beta = 1000$ less than half of one per cent. The other line is what the angle gives back: a launch at 45° instead of the best angle keeps 99.2 per cent of the best range at $\beta = 1$, 94.7 per cent at 10 and 84.3 per cent at 1000.

So at the speeds where most sports happen — $\beta$ of one to three — the air takes about half the range and the correct angle wins back about one per cent of what is left. The angle is the right answer to a question whose stakes are small; the drag is the large effect. That is worth saying plainly because the angle is what gets argued about, and the terminal speed — which decides both — is what nobody measures.

Leaving low, landing steep #

The asymmetry of the best flight is easiest to see in its two end angles. In a vacuum the landing angle equals the launch angle and both are 45°. With drag they part in opposite directions and never come back together: at $\beta = 1$ the best flight leaves at 41.0° and lands at 53.3°; at $\beta = 1000$ it leaves at 20.3° and lands at 74.3°.

A ball dropping at 74° is falling nearly at its terminal speed and nearly straight down, which is what an outfielder judging a long fly ball sees, and what a badminton player under a high clear sees even more: the shuttle arrives almost vertically wherever it was hit from. The descent angle is set by the air; the best launch adapts to it.

Where the range goes #

Plotted in drag lengths, the best range has two regimes. Below $\beta$ of about one it follows the vacuum range, $\beta$ drag lengths, and the drag is a correction. Above $\beta$ of about three it turns over and becomes nearly a straight line on a logarithmic axis: from $\beta = 100$ to $\beta = 1000$ the best range grows from 2.90 to 4.10 drag lengths, 1.2 drag lengths for a tenfold increase in launch energy. The reason is the deceleration law. A ball well above its terminal speed loses speed exponentially in the distance it covers, at a rate of one $e$-fold per drag length, so multiplying its launch speed by $\sqrt{10}$ buys the distance over which it would have lost a factor of $\sqrt{10}$ — a little over one drag length, whatever the starting speed.

This is the shuttlecock’s whole character. With a drag length of 4.7 metres, a clear hit at 30 metres a second flies about 9.6 metres at its best angle, and a stroke at a hundred metres a second, if it were hit upwards at its own best angle, would reach about 15.6: the court is a few drag lengths long, and no stroke a player can make takes the shuttle more than a few drag lengths. The game is designed around a projectile whose range is logarithmic in effort.

Height does more than the air #

The heaviest objects thrown for distance are the ones for which drag does least, and for them a different effect dominates. A ball released above the ground at height $h$ has an extra fall to make, and in a vacuum the best angle becomes $\arctan\!\big(1/\sqrt{1 + 2gh/U^2}\big)$, below 45° for any positive height. A shot released 2.1 metres up at 14 metres a second sits at $h/(U^2/g) = 0.105$ on the figure, and the height alone takes its best angle to 42.4°. Its drag, at $\beta = 0.009$, moves the angle by a twentieth of a degree.

Measured release angles in elite shot putting are lower still, 35° to 39°, and the reason is outside this model altogether: a putter can push harder along a flatter line, so the launch speed itself falls as the angle rises, and the best angle for a human is a compromise between the trajectory’s preference and the body’s. The calculation here holds $U$ fixed and says only what the flight prefers.

How the flights were computed, and checked #

Each flight is the reduced equation above, integrated by a fourth-order Runge–Kutta step until the ball passes through the ground, where the landing is found by a cubic interpolation in time that uses the velocity at both ends of the last step. The best angle is found by golden-section search on the range. Four checks stand under the figures. At $\beta = 10^{-5}$ the best angle is 44.9999° and the best range 0.999992 of $U^2/g$, which is Galileo’s answer to the precision the search was asked for. The work done by the drag along a flight, integrated alongside it, equals the kinetic and potential energy lost to five parts in ten thousand million. The range of a fixed launch with 600 steps agrees with the range at 4,800 to five parts in a million million, and the error falls sixteen-fold with each halving of the step, as a fourth-order method’s should. And the same flight integrated in metres and seconds, with a gravity and a drag length given, for balls with drag lengths of fifty metres and of two kilometres, lands at the reduced range to two parts in a hundred thousand million — which is the scaling argument above checked by not using it.

The convention: a fixed coefficient, level ground, still air #

Every number here holds the drag coefficient constant along the flight, which is what defines $L$ and makes $\beta$ the only number. A ball whose flight crosses the drag crisis — a smooth ball launched just above it, slowing through it — has a drag length that changes by a factor of four mid-flight, and its best angle is not on this curve. The ground is level and the landing point is at launch height except where a launch height is stated. The air is still: a head wind is a different problem, with a different best angle, and a tail wind another. The ball does not spin, which is the assumption the last section below removes.

What the picture cannot show #

The curve gives the angle a flight prefers and says nothing about whether a player can produce it. Launch speed and launch angle are not independent for a body — the shot put above is the clean example, and a kicked football is another, since a higher kick puts less of the foot’s speed into the ball. The figures also cannot show the drag coefficient itself, which is borrowed: 0.47 for the shot, 0.35 for the baseball, 0.25 for the football and the golf ball, 0.55 for the tennis ball’s fuzz, 0.45 for table tennis. Each moves its ball along the $\beta$ axis by the ratio of the coefficients, and the curve is so flat that a twenty per cent error in any of them moves its best angle by less than a degree. What none of them can be is a measurement this calculation made.

Who found it, and when #

Tartaglia argued in 1537 that the best angle for a cannon was 45°, before anyone had the mechanics to prove it; Galileo proved it in 1638 for a projectile with no air. Newton knew the resisted case was harder and that it lowered the range; Euler reduced the flight with quadratic drag to a quadrature in 1753, and the artillery tables computed by his method in the decades after made the first quantitative case that the best angle falls below 45°. Gunners had known it for a century from practice. The reduction to one number, the launch speed in terminal speeds, is the modern statement of Euler’s result, and it is what lets a badminton shuttle and a long fly ball be read from one figure. The same constant-coefficient drag law is what a particle is a low-pass filter replaces with Stokes’s linear one, where the corresponding length is the particle’s stopping distance and the shape of a flight is set by the Stokes number instead.

Still open: what a spinning ball’s lift does to the angle #

Every flight here has drag and gravity and nothing else, which is how a knuckleball or an unspun shot behaves and how almost no struck ball does. A golf ball leaves the club with a backspin of a few thousand revolutions a minute, a baseball off the bat with one or two, and a ball’s spin outlasts its flight, so the Magnus lift is present from launch to landing. Lift is a force across the path, growing with the square of the speed like the drag, and its job is the job the launch angle was doing — buying height. The next calculation adds it as a lift-to-drag ratio of the ball and asks how far it moves the best angle: whether a spinning ball’s best launch is a little below the curve here, or far below it, and whether there is a lift above which the best flight leaves the ground level.