The crack at the end of a whip

The pile that lands heavier than it weighs took a chain folded in half, held by both ends, and released one end. Two accounts of the fold disagreed. If links arriving at the fold lose their energy, as links landing on a pile do, the free end falls at exactly $g$; if the fold loses nothing, the falling branch carries the released energy in less and less chain, and its end accelerates past $g$, without limit in the ideal. Experiments favour the second. Real chains arrive early.

A whip is that fold lying on its side, with the weight taken away and the arm put in its place. The essay on the falling chain found that an energy-conserving fold concentrates speed; this one follows the concentration to its end, which is the only place in everyday life where a hand-held object breaks the sound barrier, and finds that what limits it is not the arm, nor the whip’s length, nor even its taper, but the mass of the scrap of string tied to its tip.

A fold on its side #

A stockwhip or a bullwhip is cracked by a throw that is mostly the shoulder and the forearm: the handle is swung forward and then stopped or pulled back, and the whip, lying out behind the hand, is thrown into a loop that runs along it towards the tip. Behind the loop, between it and the handle, the whip has come to rest, or nearly; beyond the loop, between it and the tip, the whip is moving. The loop is a fold, exactly as in the hanging chain, and it travels outward, transferring whip from the moving branch to the stationary one.

If the fold dissipates nothing, the energy the arm put into the whip stays in the moving branch, whatever happens to the branch’s mass. The moving branch’s speed is then fixed by two numbers alone:

$$v = \sqrt{\frac{2E}{M(x)}},$$

where $E$ is the energy and $M(x)$ the mass of the whip still beyond the loop when the loop has reached a position x. At the start the whole whip is moving, all 250 grams of it in the figures, and 15 joules gives it eleven metres a second — a brisk swing of the handle. As the loop runs out, the moving mass shrinks, and the speed rises as the inverse square root of what is left. When the loop reaches the tip, nothing is left moving but the tip itself.

The same arithmetic runs in the collisions that count the digits of pi, where a heavy block driving a light one against a wall can give the light one any speed at all, and in the stacked balls whose top one leaves at many times the speed they fell at. In each, kinetic energy is handed from a heavy carrier to a light one without loss, and the light one has to move fast to hold it.

Why energy and not momentum #

The calculation leans entirely on the fold keeping its energy, and the whip is the best everyday evidence that a fold can. Suppose instead the fold behaved like the pile of chain on a scale, where each link arriving is stopped dead and its kinetic energy lost. Then a link passing through the loop would arrive at rest on the stationary side by an inelastic collision, and the moving branch would lose energy at exactly the rate needed to keep its speed constant. In the falling chain that dissipative account gives a free end that falls at $g$ and no faster. In the whip it gives a moving branch that never speeds up at all: the loop would run out to the tip at the speed the arm gave it, and the tip would arrive at eleven metres a second. No whip would ever crack.

Whips crack, so the fold in a whip must keep most of its energy. The reason is the same one the falling chain suggested: a plaited leather whip, like a smooth cord, has no rigid links knocking against one another at the fold, and the material rolls round the loop rather than being brought to rest by impact. The spring whose bottom does not know it has been dropped met the opposite case, a collapsing front in which the coils do collide and the energy goes into the collisions. The whip is on the other side of that line, and the crack is the sound of being there.

Mass leaving, energy staying #

The second figure follows the moving mass. For a uniform whip it falls in proportion to the length left. Half-way along, half of the whip is still moving; ten centimetres from the tip, a twentieth. A whip that tapers from its handle to its tip loses mass faster, because the part left to move is the thinnest part: a linearly tapered whip of the same total weight has a quarter of its mass beyond the half-way point, and ten centimetres from the tip just over a gram is moving.

That difference shapes the first figure. In the tapered whips the speed builds steadily, decade by decade of remaining length, so that half a metre from the tip they are already moving at fifty to seventy metres a second. The uniform whip stays slow for most of its length and then rises abruptly at the very end; it passes the speed of sound only in its last millimetre, where the tapered whips cross a centimetre and a half from the tip.

What the figure also shows, and what is easy to miss, is that all three curves end in the same place. When the loop reaches the tip, the moving mass is the tip itself, and in all three whips the tip is the same 0.2-gram cracker. In this model the end’s speed does not depend on the taper at all.

The ceiling a scrap of string sets #

The ceiling is $\sqrt{2E/m_{\text{tip}}}$, and the third figure draws it. To move faster than sound, 343 metres a second in air at room temperature, the tip must be lighter than $2E/c^2$. For a throw of fifteen joules that is a quarter of a gram, which is why the end of a whip carries a cracker or popper: a short tuft of string, horsehair or nylon, light, flexible and replaceable. A whip tied off with a heavy knot, or one whose cracker has worn away to leave the thick leather of the fall, does not crack however hard it is thrown. A heavier throw buys a heavier cracker, but only as the square root: forty joules allows two-thirds of a gram.

The arithmetic is easier to believe for a towel. A wet towel, flicked, cracks too, and a towel is not tapered. What it has is a corner — a little mass at the end of a large one — and the energy given to the whole towel by the wrist ends up in the corner. The taper of a whip is a refinement of the towel’s corner, not its replacement.

What the taper is for #

If the taper does not set the final speed, it does something else. A real whip is not a perfect, lossless fold. The moving part pushes air aside as it goes, bends and unbends, rubs its own plaits against one another, and every one of those losses takes energy from the moving branch along the way. A whip in which the speed builds gradually, from the handle out, keeps its moving part compact and its loop well formed; a uniform rope tends to let the loop spread and stall before it reaches the end, which is why a heavy uniform rope, thrown the same way, rarely cracks even when its end is light.

How much loss a whip can afford follows from the same formula.

The fourth figure takes the tapered whip and lets the moving branch lose a fixed fraction of its energy for every metre the loop travels. The speed goes only as the square root of the energy, so a loss of a fifth of the energy costs a tenth of the speed. But a whip whose cracker would carry it just past the sound barrier with no loss at all has no tenth to spare. With a 0.2-gram cracker on a fifteen-joule throw the margin is about twenty per cent of the energy over the whole two metres, and beyond that the whip fails to crack. A lighter cracker or a harder throw buys margin; so does a shorter whip, which gives the losses less length to act on. A whip-maker’s choices of length, weight and taper are a trade among those.

A whip that is designed not to crack #

Fly-fishermen cast the same way. A fly line is thrown forward, and a loop runs out along it, unrolling the line towards the fly, exactly as a loop runs out along a whip. The line is tapered too, from a heavy belly to a fine leader, and for the same reason: so that the loop carries its energy smoothly out to the end and lays the fly down at the far end of the cast. But an angler wants the opposite of a crack. A cast that snaps — the loop accelerating the leader’s end past the speed the fly can survive — whips the fly off the line, and the crack of a badly timed cast is a familiar sound on rivers.

The difference between the two instruments is the arithmetic of the third figure, read in reverse. The fly at the end of a leader, with its hook and hackle, is heavier than a whip’s cracker, and the energy a gentle forward cast gives the line is small, so $\sqrt{2E/m}$ stays far below the speed of sound. A caster who throws too hard, or whose leader tapers too finely to a light end, moves towards the whip’s regime. The same taper that lets a whip build speed gradually lets a line deliver its energy gently, and what decides which happens is how much energy arrives and how much mass is waiting for it at the end.

A slow throw that ends in a flick #

The last figure turns position into time. A fold travels at half the speed of its moving branch — the branch pays out whip at its own speed, and the fold sits half-way between the moving and stationary parts — so the loop’s journey can be timed. It takes 185 milliseconds to run the two metres, which matches the experience of cracking a whip: a visible, unhurried roll that runs out along the whip and ends in a crack. Almost all of that time is spent slow. The speed reaches a hundred metres a second only in the last two milliseconds, and passes the speed of sound eighty microseconds before the end.

This is why the physics was so hard to see. The crack was long thought to be the sound of the whip slapping against itself, until high-speed photographs, beginning with shadowgraphs in the 1920s, caught the tip moving faster than sound and trailing the shock wave that makes the sound. The event that produces it lasts tens of microseconds and happens within a few centimetres of the tip, at the end of a motion that took a fifth of a second.

The crack itself #

An object moving faster than sound outruns the pressure waves it makes. They pile up along a cone, the shape the cone the source leaves behind found trailing any source that moves faster than the waves it emits, and the cone is a shock: a sudden jump in pressure that, as the front that steepens until it cannot showed for any wave strong enough, has nothing smooth about it. Swept past an ear, it is a crack. A whip’s crack is a sonic boom of the same kind an aircraft makes, from a source a few millimetres across, which is why it is short and sharp rather than a double thump.

Exactly what part of the whip makes the boom has been argued. The tip goes supersonic, but so, in the final instant, does part of the loop, and some careful measurements in the 1990s and 2000s have suggested the loop or the tuft unravelling at the end makes the loudest shock. The energy argument does not depend on which: it says only that something at the end must move faster than sound, and that a light enough end will.

Where the argument holds #

The calculation assumes the fold is lossless, then adds loss by hand at a fixed rate. Neither is exactly true. The fold in a plaited whip loses some energy at every bend; the loss to the air grows as the cube of the speed, so it is concentrated at the end, where the speeds are high but the moving part is small; and the arm does not deliver its energy and stop, but goes on pulling on the handle during the early part of the throw. The figures also treat the whip as perfectly flexible, which a leather whip nearly is, and the loop as a sharp fold, which it is not quite: a real loop has a radius set by the whip’s stiffness and its speed, and the stiffness matters most where the whip is thickest.

The numbers used — a two-metre whip of 250 grams, fifteen joules from the arm and a 0.2-gram cracker — are of the order of a stock whip and a good throw, chosen to show the effect, and the conclusion does not depend on them in detail: change any of them by a factor of two and the ceiling moves by a factor of about 1.4.

What the figures cannot show #

The figures follow a single number, the speed of the moving branch. They cannot show the shape of the loop, how it tightens as it travels, or the way the whip’s tip whips round the loop’s last turn. They cannot show where the loop’s own energy goes, which in the fold model is all with the moving branch but in a real whip is partly in the loop’s curvature and its rotation. And they cannot show the sound. A whip’s crack is loud enough at close range to be a hazard to hearing, from an energy budget of a few joules — but the share of the energy that becomes sound is tiny, and computing it needs the detailed motion of the tip that the figures replace with a single speed.

Still open: what exactly makes the boom #

Whether the loudest shock comes from the tip, from the loop as it passes the tip, or from the cracker’s fibres splaying apart in the last instant is still investigated with high-speed cameras, schlieren photography and microphones placed around the path. The answer seems to differ between whips and throws. Models of the whip as an elastic rod with a taper, solved in full rather than reduced to an energy balance, reproduce the acceleration of the loop and predict the tip speed, and they have been used to design whips that crack more easily. Whether a single account covers the bullwhip, the stockwhip, the towel and the cracking rope — all of which crack, with very different shapes — and what fraction of the arm’s energy each delivers to the tip, are questions answered experiment by experiment rather than in general.

The habit worth carrying away is to follow energy into the smallest place it can go. A whip is an energy-conserving fold: the 15 J the arm gives it stays in the part still moving, whose mass falls from 250 g to the 0.2 g of the cracker as the loop runs out, so its speed $\sqrt{2E/M}$ climbs from 11 m/s at the handle to 387 m/s at the tip — past sound if the cracker weighs less than 2E/c², a quarter of a gram — with the taper deciding how gradually the speed builds and the cracker deciding how high it goes. The crack is not made by the arm, which is slow, but by the fact that the end is light.