The ball thrown up that lands to the west

The angle that throws furthest found the forty-five degrees of a throw in a vacuum over a flat, still ground, and everywhere a throw can reach drew the envelope of all such throws. The angle that drag moves added air, and the throw most likely to go in the scatter of a real hand. Each of those assumed the ground stands still. It does not: the forces that are not there found that anything moving over the turning Earth is pushed sideways by the Coriolis force, and the deflection that closes on itself followed a body moving freely over the Earth’s surface round an inertial circle.

Both of those concerned horizontal motion, the part of the Coriolis force that turns winds and currents to the right in the northern hemisphere and vanishes at the equator. There is a second part, which acts on vertical motion, is largest at the equator, and vanishes at the poles. It is small — millimetres for anything a person can throw — and it has a feature that makes it worth an essay: a ball thrown straight up and a ball dropped from the same height are deflected in opposite directions, and the thrown one four times as far. The explanation turns a problem about rotating frames into a problem about the area under a graph.

Why height makes a difference #

At latitude $\lambda$ the Earth’s surface moves east at $\Omega R\cos\lambda$, where $\Omega$ is the Earth’s rotation rate and $R$ its radius; a point a height $h$ above the surface moves east a little faster, at $\Omega(R + h)\cos\lambda$, because it is further from the axis. A ball held at the top of a tower therefore moves east faster than the ground at the tower’s foot, by $\Omega h\cos\lambda$ — about five millimetres a second for a hundred-metre tower at forty-five degrees. Let it go and it keeps that extra eastward speed as it falls, while the ground below moves a little slower. It lands east of the point directly beneath.

The same reasoning run the other way explains the throw. A ball thrown straight up leaves the hand with the eastward speed of the hand, and as it rises it passes through air, and over ground, whose “correct” eastward speed at its height is larger. It is behind. It drifts west relative to everything at its height, and it keeps drifting west on the way down. The deeper reason is the quantity that survives a change of shape: a body moving away from an axis conserves its angular momentum about it, so its angular velocity falls as its distance grows — the same reason a spinning skater slows with arms outstretched.

One line of algebra #

In the frame of the ground, with $x$ pointing east and $z$ up, the Coriolis acceleration on a body moving vertically at speed $v_z$ is $-2\Omega\cos\lambda\,v_z$ in the east–west direction. The vertical speed is the rate of change of height, so this integrates at once:

$$v_x = -2\Omega\cos\lambda\,(z - z_0).$$

The east–west velocity of any body moving vertically is proportional to its height above the point where it started, with a minus sign — westward above the start, eastward below it. Integrating again, the total east–west displacement is $-2\Omega\cos\lambda$ times the integral of $z - z_0$ over the flight: the area between the body’s height–time curve and its starting height.

The figure turns the asymmetry into geometry. The dropped ball starts at a hundred metres and falls; its height–time curve lies below its starting height, and the area between them — the upper shaded region — is a third of the height times the fall time, because a body falling from rest spends most of its fall near the top. The thrown ball starts on the ground and rises; its curve lies above its starting height for twice as long, and the area beneath it — the lower shaded region — is four-thirds of the height times the fall time, the area under a parabola. One area is below the start and one above, so the drifts have opposite signs; and one is four times the other, so the throw lands four times as far away.

The two paths #

Drawn with the horizontal scale magnified two thousand times, the two paths look like this. The dropped ball curves steadily eastward, slowly at first and faster as it falls, landing fifteen and a half millimetres east. The thrown ball moves west on the way up — fastest when it is highest, since its westward speed is proportional to its height — and continues west on the way down, slowing its westward drift as it descends, until it lands sixty-two millimetres west of the hand. It never moves east at any point in its flight. The instinct that it should retrace its path, because it rises and falls through the same air, is wrong because the Coriolis force depends on the direction of motion: rising and falling push the ball in opposite directions, but the westward velocity it has gained on the way up is not undone until it is back at its starting height, and by then it has been moving west for the whole flight.

The area is Kepler’s #

The area rule has a second derivation that shows where it comes from, and it needs no rotating frame at all. Seen from space, a ball released from a tower at the equator is not falling straight down: it is in orbit round the Earth’s centre, on a very narrow ellipse that happens to intersect the ground. Like every orbit it conserves angular momentum about the centre, so its angular speed round the Earth’s axis varies inversely as the square of its distance from the centre — Kepler’s second law, equal areas in equal times.

It starts with the angular speed of the tower’s top. As it falls, its distance from the centre shrinks, and its angular speed rises above the Earth’s; it gets ahead of the ground turning beneath it, and lands east. The ball thrown upward starts with the angular speed of the ground; as it rises its distance from the centre grows and its angular speed drops below the Earth’s, so it falls behind; and on the way down it only recovers the Earth’s rate when it is back at the ground, so it has been behind for its whole flight and lands west. The lag in angle accumulated over the flight is the integral of the difference between the two angular speeds, which for a small height is $2\Omega$ times the height divided by the Earth’s radius; multiplied by the radius to turn an angle into a distance, it is $2\Omega$ times the integral of the height over time. The area under the height–time graph is Kepler’s area in disguise.

A throw that lands straight below #

The area rule makes one more prediction that is pleasant to check. A ball thrown upward from the top of a tower spends part of its flight above its starting height and part below it, on the way to the ground. The part above pushes it west and the part below east, and the two can cancel. The signed area vanishes when the ball is thrown up at $\sqrt{2gh/3}$, which carries it a third of the tower’s height above the top before it falls. A ball thrown upward from the top of a hundred-metre tower at 25.6 metres a second rises 33 metres, falls 133, and — alone of all the ways of releasing it — lands exactly at the foot of the plumb line from where it left the hand. Every slower throw lands east of that point and every faster one west, and the result is the same at every latitude and in every gravity, because it depends only on the shape of a parabola.

A measurement in a mine shaft #

The eastward deflection of a falling body was one of the first proposed tests of the Earth’s rotation, suggested by Newton in a letter to Robert Hooke in 1679, in an exchange that went on to argue about the shape of the path a body would follow if it could fall through the Earth. Attempts to measure it from church towers in Bologna and Hamburg around 1800 found eastward deflections of roughly the right size and spurious southward ones, and were bedevilled by air currents and by the difficulty of releasing a ball without giving it a push.

The deflection grows as the height to the power three-halves — as the cube of the fall time — so the experiment wants the deepest available drop. In 1831 Ferdinand Reich used a mine shaft at Freiberg in Saxony, 158.5 metres deep, where the air was still, and dropped more than a hundred balls. Their average landing point was 28.4 millimetres east of the plumb line, against 27.6 predicted by the formula here at his latitude. The scatter of individual drops was many times the deflection itself, and the agreement was a triumph of averaging: the effect is real, it is the size the rotation predicts, and it needs a hundred-metre drop to reach the size of a thumbnail.

The westward landing of a thrown body is four times larger and much harder to measure, because nobody can throw a ball exactly vertically. An error in the direction of the throw of one part in ten thousand moves the landing point by more than the whole effect. The corresponding effect on things launched upward on purpose — sounding rockets, and instruments carried up by balloons and dropped — is a routine correction in their tracking, but a clean demonstration with a ball and a hand is out of reach.

Strongest at the equator #

The vertical Coriolis effect goes as the cosine of the latitude: largest at the equator, zero at the poles, the reverse of the horizontal effect that turns winds. The reason is visible in the first explanation. At the equator a vertical line points straight away from the Earth’s axis, and moving along it changes a body’s distance from the axis as much as possible. At a pole a vertical line is the axis, and moving up or down it changes nothing — the ground turns but carries nothing eastward. The two Coriolis effects are the two components of one vector, the Earth’s angular velocity, resolved along the local vertical and the local north; one acts on horizontal motion and one on vertical motion, and between them they share out the rotation according to latitude.

Shells that land long and short #

The vertical effect also acts on horizontal motion through a less obvious route: a body moving east is moving round the Earth’s axis faster than the ground, so it feels a slightly larger centrifugal effect — it is lighter — and a body moving west is heavier. This is the Eötvös effect, measured by Loránd Eötvös in 1908 as a change in the weight of objects carried east and west on ships, and it is the same vertical Coriolis term, $2\Omega\cos\lambda$ times the eastward speed, acting upward.

For a long-range shell both effects matter, and the figure computes them for a shell fired at 800 metres a second and forty-five degrees elevation, in a vacuum over a flat Earth. Every shell lands to the right of its aim, by several hundred metres: that is the horizontal part, the one that turns winds. Fired east, the shell is lighter throughout its flight and carries 257 metres further; fired west, it is heavier and falls 260 metres short. Fired north or south, the range hardly changes, but the rightward drift is different for the two, 259 metres going north and 517 going south, because the shell’s own rise and fall carries it west, exactly as the ball thrown upward is carried west — and west is to the left when firing north and to the right when firing south.

The flat Earth and the vacuum are both wrong for a sixty-five-kilometre shot, and a real shell travels less far and for less time; the numbers illustrate the size, not a firing table. The pattern — always right of aim in the northern hemisphere, long to the east, short to the west, and the vertical drift adding to or taking from the sideways one — is what artillery tables have corrected for since guns reached tens of kilometres, and what long-range marksmen at a kilometre, where the effects are a few centimetres, correct for now.

The term the weather leaves out #

The same term appears, on a much larger scale, in the equations for the atmosphere and oceans — and it is usually thrown away. Weather and climate models are built on what meteorologists call the traditional approximation, which keeps the Coriolis force on horizontal motion, the part proportional to the sine of the latitude, and drops the part studied here, proportional to the cosine: the westward push on rising air, the eastward push on sinking air, and the Eötvös lightening of eastward winds. The justification is geometric. The atmosphere and the oceans are thin sheets, a few kilometres deep over a planet thousands of kilometres across, so their vertical motions are thousands of times slower than their horizontal ones, and a term proportional to vertical velocity is small.

The justification fails in exactly the places where the cosine is largest and the vertical motion strongest. Near the equator the horizontal Coriolis force, which goes as the sine of the latitude, vanishes, and the neglected term is then the only rotational effect left. In deep convection — the plumes of dense water that sink kilometres in the polar and subpolar oceans in winter, or the towering updraughts of tropical thunderstorms — vertical velocities are no longer small, and the neglected term tilts the plumes and changes how they mix. Studies of these “non-traditional” effects find that they alter the structure of equatorial currents and deep convective plumes measurably, and whether climate models that omit them are missing something that matters for the large-scale circulation is an active question. A ball thrown upward on a still day is the smallest instance of a force the ocean may be using on a scale of kilometres.

Where the calculation stops #

The calculation keeps only the first order in the Earth’s rotation rate, which is ample: the next order is smaller by a factor of the rotation rate times the flight time, a few parts in a hundred thousand for any flight shorter than a minute. It treats gravity as uniform and vertical, which ignores the small southward component of a dropped body’s deflection in the northern hemisphere, a second-order effect of the Earth’s centrifugal flattening of the direction of gravity. For long flights the curvature of the Earth changes the direction of “down” along the path, and the flat-Earth shell calculation above is an approximation for exactly that reason. And air — the thing that defeated the tower experiments — dominates everything: a ball dropped a hundred metres through air reaches a terminal speed that shortens its flight and changes its area, and a light breeze deflects it by far more than the rotation does.

What the pictures cannot show #

The figures show the deflection in the ground’s frame, where it is produced by a force that is not there. In a frame that does not rotate there is no force at all; the ball simply moves in a straight line horizontally, at the speed the tower’s top gave it, while the tower’s foot is carried round a circle by the Earth, and the deflection is the difference between a straight line and an arc. Both descriptions give the same landing point, and the figures cannot show which is “really” happening, because nothing in the physics prefers one. They also cannot show the hardest part of the measurement: that a ball must be let go without the slightest sideways push, a tolerance of a micrometre a second, far smaller than any hand or clamp naturally achieves.

Still open: what the Earth’s rotation does to things too small to feel it #

The vertical Coriolis effect is too small to matter for anything a person throws, and it is now one of the most precisely measured forces in physics — in atom interferometers, where clouds of atoms thrown upward and let fall are split and recombined, and where the phase difference picks up exactly the rotation term computed here, proportional to the area enclosed by the two paths. In those instruments the Earth’s rotation is a nuisance to be subtracted, several orders of magnitude larger than the gravitational effects being sought, and how completely it can be removed — by rotating the instrument against the Earth, by symmetric launches, by measuring it and correcting it — sets the accuracy of some of the most sensitive tests of gravity. Whether atom-interferometric measurements of rotation will eventually compete with the best ring-laser gyroscopes, which detect rotation through the Sagnac effect, is an open question in inertial sensing.

The habit worth carrying away is to look for a quantity that a force depends on, and integrate it. For anything moving vertically on the turning Earth the sideways velocity is proportional to its height above its start, so its drift is proportional to the area under its height–time graph — which is why a ball thrown up to a height lands four times as far west as one dropped from it lands east. The Coriolis force became a matter of reading an area off a parabola.