Concept

Trajectory — where it appears

The path a particle or a body actually takes through space, as against the streamlines of the field it moves in. It is an integral of the forces along the way, so two bodies launched identically into different histories arrive in different places.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

Four turns of a wave, and the parcel is not back. Two parcels traced through four periods of a linear deep-water wave of steepness 0.1, by integrating the exact velocity field. Each orbit is very nearly a closed circle and misses closing by a little, every time, in the same direction — that miss is the whole of the Stokes drift. The near-surface parcel advances 0.0357 of a wavelength over the four cycles and the one a tenth of a wavelength down advances 0.0122, a third as far — because the drift falls off twice as fast with depth as the orbit's own size does.

The drift in a wave that has none

The velocity at any fixed point under a passing wave averages to exactly zero, and every parcel of water in it moves steadily forward anyway. The orbits do not close, they miss by the same amount every time, and the missing amount is the square of the steepness times the wave speed.

kinematics · Stokes drift

Five points, and where each one's fluid came from. Back-trajectories through six units of time in an unsteady double gyre. Each curve ends at the place the fluid now at the marked point started; nothing about that place can be read off the velocity at the marker.

A scalar is a record of where its fluid was

A conserved scalar has no value of its own. Its value at a point is whatever it was at the place that point's fluid started from, which makes a dye field a photograph of the past — and makes the map from now to then the only thing in the flow that carries the past at all.

kinematics · Advection $Spin against distance flown, not against time. The spin of a struck golf ball as a fraction of its launch spin, against how far it has flown. The integrated flight sits exactly on an exponential in distance, because the spin-down torque is proportional to speed times spin and the speed cancels. The constant is 1,202 metres and the drive is 200, so the ball arrives with five sixths of the spin it left with.$

The ball that never forgets its spin

Commentary explains a late-swerving ball by saying the spin is dying away. The spin-down torque goes as speed times spin, so spin decays over a distance rather than a time — 1,202 metres for a golf ball against a 200-metre drive — and what dies away is the speed, which makes the swerve stronger.

applied · Sport ball

The line the dye actually draws

The streakline is the curve most photographs really show, and of the three curves it is the hardest to compute, because it needs the whole history of the flow. This draws it, in a flow where all four curves have closed forms, and the third curve turns out to be a different kind of object from the other two rather than a third example of the same one.

kinematics · Flow curves

Every ball's best angle is on one curve. The launch angle that carries a ball furthest over level ground, against β, the square of its launch speed measured in terminal speeds. At small β the air does nothing and the answer is 45°. It falls slowly as β grows: 41° at β = 1, 32.6° at 10 and 20.3° at 1000. A shot put sits at β = 0.009 and 44.9°, a golf ball struck without spin at β = 2.39 and 38.1°, a shuttlecock hit at 30 m/s at β = 19.5 and 30.2°.

The terminal speed sets the best launch angle

In a vacuum every ball goes furthest at forty-five degrees. In air the best angle is lower, and it is lower by an amount that depends on one number only: how many times its own terminal speed the ball is launched at. A shot put, a baseball and a shuttlecock are three points on one curve, which falls from 45° to 41° at once and then needs a thousandfold more speed to reach 20°.

applied · Sport ball $Lift takes the launch angle down far faster than drag does. The best launch angle against β for a ball whose lift is a fixed fraction ℓ of its drag, from ℓ = 0 — the drag-only curve — to ℓ = 1. At β = 10 drag alone puts the best angle at 32.6°; a lift-to-drag ratio of 0.4 takes it to 15.9° and 0.8 makes a level launch best. Once the curve reaches zero it stays there: the lift lifts the ball, and any angle given at launch is height the lift would have bought for less drag.$

A spinning ball should leave low

Drag alone takes a ball's best launch angle from 45° to the high thirties at the speeds sport is played at. Backspin takes it into the teens, because a ball with lift is a glider and a glider buys its height with lift rather than with angle. Every tenth of lift-to-drag ratio takes a fixed slice off the angle, and above a line that depends on the launch speed the best flight leaves the ground level.

applied · Sport ball

With lift in proportion to spin, a drive gains from spin until its launch is level; the measured curves give it a best spin where drives are hit. The longest carry of a 70 m/s drive at each spin rate, its launch angle chosen for that spin, for four pairs of coefficients: lift in proportion to the spin ratio with fixed drag; drag rising with spin; lift growing as its 0.4 power; both measured forms together. The fixed pair peaks near 4,800 rpm, where its best launch reaches level, and drag rising alone near 4,400 rpm; lift saturating alone is still rising at 6,000; with both measured forms the carry peaks at 234.6 m at 2273 rpm and 16.8°.

A drive's best spin falls where players hit it only because lift tires and drag grows

With lift in proportion to spin and drag fixed, a golf drive keeps gaining from backspin until its best launch has fallen to level, near 4,800 rpm, and past a few thousand more the model flies a ball the length of eight football pitches. Give lift and drag the curves wind tunnels measure — lift growing ever more slowly with the spin ratio, drag rising with it — and the best spin moves to near 2,300 rpm at 70 m/s, with a launch well above level, inside the range launch monitors recommend. Either curve alone leaves it at twice that or more. And along a ridge of best launches the carry hardly moves while the angle falls by ten degrees, which is why the launch angle is the softer of the two numbers.

applied · Sport ball

Along the carry's ridge the run hardly moves the best spin, because every drive on it lands alike. The total distance of a 70 m/s drive, carry and run, at each spin rate launched at the angle that carries furthest for that spin, on a firm, a medium and a soft fairway, with the carry alone. The carry peaks at 234.6 m at 2273 rpm; the total along the ridge peaks at 1728 rpm on the firm turf, 274.3 m, 2052 rpm on the medium turf, 249.3 m, 2375 rpm on the soft turf, 239.9 m. Every drive on the ridge lands between 45.2° and 49.2°.

The run moves the best drive below the ridge that carries furthest

A drive's carry is greatest along a ridge of launches on which more spin trades for a lower angle at almost no cost. Every drive on that ridge lands at about the same steep angle, so the run after landing hardly chooses between them. What the run rewards is leaving the ridge: less spin and a launch several degrees lower, a lower flight that lands flatter, gives up thirteen metres of carry and gets back twenty in the run. On an ordinary fairway the best drive in all lands near thirty-five degrees, at every ball speed; on a firm one the model's ground stops being a turf anybody has measured before the run stops paying.

applied · Sport ball

Named alongside it

The objects these essays reach for when they reach for this one.

Magnus effect Model limit Optimisation Spin Dimensional analysis Drag Lift coefficient Measurement Angular momentum Drag coefficient Flow map Initial condition

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