The jump that does not get higher with size

The floor that does no work followed the energy of a jump to its source. The floor pushes the jumper up but does no work, because the patch of floor under the foot never moves; the three hundred joules a person leaves the ground with come from the chemistry of their own muscles. The push that pays only for the hill found a line of dominoes growing by a fixed factor at each step, because each one held energy stored when it was stood up and the push from its neighbour paid only to release it.

This essay puts those two ideas together and asks a question that sounds as though it has nothing to do with either: how high can an animal jump, and how does that depend on its size? The answer has a part that is almost too simple to believe and a part that explains the strangest mechanisms in insect legs. The simple part says that jump height does not depend on size at all. The other part says why the smallest animals cannot reach that height with muscle, and how every one of them does it with a spring instead.

Borelli’s cancellation #

Giovanni Alfonso Borelli, in a book on animal motion published in 1680, the year after his death, made the argument with nothing but proportion. Suppose an animal’s jumping muscles are a fixed fraction of its body, and that each kilogram of muscle can do a fixed amount of work in one contraction. Then the work available for a jump is proportional to the animal’s mass. The work needed to raise the animal’s centre of mass by a height h is its weight times h, also proportional to its mass. The mass cancels, and the height is the same for every animal built to the same plan.

The numbers support it to within a factor of a few across a range of sizes that makes the agreement remarkable. A person standing still and jumping raises their centre of mass about half a metre. A locust, two grams, takes off at about three metres a second, which in vacuum would lift it half a metre. A flea, less than half a milligram, leaves at about two metres a second and rises eighteen centimetres. A frog, a galago and a froghopper are in the same range or a little higher. Between the flea and the person is a factor of a hundred million in mass, and the heights differ by a factor of three.

The flat line in the figure takes the jumping muscle to be a tenth of the body and to do sixty joules per kilogram, which gives a rise of 0.61 metres. Neither number is exact for any animal, and the point is not the value but the flatness. The often-repeated comparison — a flea jumps a hundred times its own length, so a flea the size of a person would clear a skyscraper — gets the scaling exactly backwards. A flea jumps a hundred body lengths because it is small. A person-sized flea, built to the same plan, would rise about half a metre like the person.

Why a kilogram of muscle does a fixed amount of work #

Borelli assumed the constant and could not explain it; the explanation came three centuries later, from the structure of muscle. A muscle fibre is a chain of identical contractile units, sarcomeres, each a couple of micrometres long, in which filaments slide past one another driven by molecular motors. Every vertebrate and insect muscle studied builds them to nearly the same pattern. Two consequences follow. The force a muscle can exert is proportional to the number of sarcomeres side by side, its cross-sectional area, and the stress it can sustain is nearly universal, about three hundred kilopascals. The distance it can shorten is proportional to the number of sarcomeres end to end, its length, and the fraction it can shorten by is nearly universal too, roughly a third.

Work is force times distance, so the work a muscle can do in one contraction is proportional to its area times its length, which is its volume, and so to its mass. With three hundred kilopascals acting through a third of the length, it comes to about a hundred kilojoules per cubic metre, a hundred joules per kilogram, at the most favourable rate of shortening and somewhat less in practice. Borelli’s constant is the product of two constants of molecular biology, and the cancellation of size in jump height is the cancellation of that product against gravity.

The time a jump takes #

The flat line is not the whole story, and the sloping line in the figure shows why. Energy is a fixed amount per kilogram, but it must be delivered while the animal is still on the ground, and that time is short. A jump is a push that lasts as long as the legs take to straighten. Starting from rest and reaching the take-off speed v over a leg extension L takes a time of about 2L/v, and since leg length scales with body length, roughly as the cube root of mass, the time available shrinks with size.

For a person the legs extend about forty centimetres and take a quarter of a second. For a locust, a couple of centimetres, seven milliseconds. For a flea, less than a millimetre, under half a millisecond. Delivering a fixed energy per kilogram in a time proportional to body length means delivering a power per kilogram inversely proportional to body length, and that is where the size dependence comes back in.

Muscle has a maximum power per kilogram, a few hundred watts, set by how fast its molecular motors can cycle and how much force each can exert. It is roughly the same in a flea’s leg as in a person’s thigh, as the speed at which grip hands over to power found for an engine, where a car’s power limits its push at speed rather than its grip. A person jumping uses close to all of it, which is why the power limit and Borelli’s limit meet near human size in the first figure. A locust would need thirty times what muscle can give, and a flea more than five hundred times. If they pushed directly with muscle, the power limit would hold them to a centimetre or two.

A spring that buys time #

The way round is not stronger muscle but a different arrangement of the same muscle. Let the muscle work slowly, for a tenth of a second, bending a spring that is held from releasing by a catch. When the spring is fully loaded the catch lets go, and the spring returns its energy to the legs in the millisecond they take to straighten. The energy is the muscle’s, every joule of it; the power at take-off is a hundred times the muscle’s, because the spring delivers in a millisecond what the muscle took a hundred to provide.

This is the domino’s lesson with time in place of energy. The push that pays only for the hill found a small push releasing a large stored energy because the energy was put in earlier, when the domino was stood up. A catapult does the same with power: the trigger that releases a flea’s jump does almost no work, and the work was done in the slow loading that preceded it.

Every small jumper whose jump has been examined closely uses such a mechanism. Henry Bennet-Clark and Edward Lucey showed in 1967 that a flea loads a pad of resilin, a rubbery protein, in its thorax, holds it with a catch and releases it in under a millisecond; resilin returns about ninety-seven per cent of the energy put into it, better than most rubbers. A locust bends stiff semi-circular plates of cuticle at its knee joints for several hundred milliseconds before a jump. The froghopper, a sap-feeding insect of about twelve milligrams, loads a bow-shaped structure of cuticle and resilin and leaves the ground at 4.7 metres a second, enduring an acceleration of about four hundred times gravity. Frogs stretch their leg tendons before the joint begins to move, and galagos, the small primates that jump more than two metres, do the same with theirs.

How much a spring can hold #

A spring that is to stand in for muscle must store at least as much energy per kilogram as the muscle delivers, and the materials animals use store far more. A tendon can be stretched by about eight per cent at a stress of a hundred megapascals before it is in danger, and the energy it stores, half the stress times the strain, is a few kilojoules per kilogram of tendon, thirty times what a kilogram of muscle does in a contraction. Resilin and the cuticle of insect legs are in the same range. A small spring therefore suffices to bank the work of a large muscle, which is why a flea’s resilin pads are a tiny fraction of its body.

The spring cannot release faster than its own mass allows. A spring unloading in a millisecond must accelerate its own coils along with whatever it pushes, and a uniform spring behaves as if a third of its mass were attached to its moving end. A spring heavy compared with the leg it drives spends its energy moving itself. The fastest biological catapults use springs that are light and stiff, and their latches are arranged so that the spring’s own motion takes as little of the release as possible.

Why a person does not need one #

The figure puts human beings close to the crossing, where muscle alone can just about deliver the energy in the time the legs take to straighten, and that is why a person can jump well without any catch. The springs are still there and still help. Tendons store energy when they are stretched, and a person who dips before jumping — the countermovement every jumper makes without thinking — loads the muscles and tendons of the legs during the dip, so that the upward push starts with the muscles already active and some energy already stored. The countermovement adds a few centimetres to a standing jump. It does not change the picture, because a person has the time the flea does not: a quarter of a second of push is long enough for muscle to work at nearly its full power.

Larger animals are on the other side of the crossing, where the energy limit binds and the power is easily available, and there the flat line should apply on its own. In fact the largest animals jump less than Borelli’s argument allows, for a reason outside it: their bones and tendons are stressed by landing in proportion to their weight per unit cross-section, which rises with size, and an elephant that jumped half a metre would break its legs coming down. The same argument from the strength of materials that limits how tall a mountain can be limits how much a large animal can afford to accelerate.

Springs serve the large too, for a different purpose: not to make one jump possible but to make many cheap. A kangaroo hopping stores the energy of each landing in the long tendons of its hind legs and returns most of it in the next take-off, so the muscles only top up what the tendons lose. Measured on a treadmill in 1973, a kangaroo’s rate of using oxygen stayed nearly level as it hopped faster over a wide range of speeds, while a running dog’s or a person’s rises steadily. The tendon is a catapult loaded by gravity instead of by a muscle and a latch, and the hop is a bounce sustained by a small payment on every cycle.

The air that small jumpers climb through #

At the small end a second limit appears, this time from outside the animal. Air drag on a moving body grows with its cross-sectional area, and its weight with its volume, so the ratio of drag to weight grows as the body shrinks, as the cube root of mass in reverse. For a person the air makes no visible difference to a jump. For a flea it takes away about a seventh of its height at the speed it actually achieves, and it would take away half if the flea could leave at five metres a second.

That puts a ceiling on the strategy of a stronger spring. A smaller jumper can store more energy per kilogram in resilin than its muscle could deliver, but it gains less height from each increment of speed, because drag grows as the square of the speed. The smallest jumping insects, a fraction of a milligram, are at the edge where a better catapult buys almost nothing; the angle that drag moves found the same exchange of speed for range in a thrown ball, much more mildly.

What a run-up adds #

The energy argument also explains the one way an animal can rise much further than Borelli’s half-metre: by arriving at the jump already moving. A long jumper takes off at about twenty degrees rather than forty-five, because a steeper take-off would cost more horizontal speed than it gains in time aloft — the trade the best throw is a tangency found. A sprinter reaches ten metres a second over a run of thirty or forty metres, building speed over many strides, each of which has a quarter of a second of push. Turned into height, that speed is worth $v^2/2g$, five metres. A pole vaulter does exactly that conversion. The pole bends as the vaulter plants it, storing the run-up’s kinetic energy as elastic energy, and straightens, returning it upward; the vaulter’s arms add a little more near the top. The rise of the centre of mass in a world-class vault, from about a metre at take-off to over six at the bar, is the run-up’s speed, banked in the pole for a second and spent as height.

The pole is the flea’s resilin at human scale, used for the opposite reason. The flea stores energy so as to deliver it faster than its muscle could. The vaulter stores energy so as to redirect it, from horizontal to vertical, which neither a body nor a floor can do without losing most of it in the turn. Both are catapults, and neither adds a joule.

What the scaling leaves out #

The lines in the figures are a scaling argument with three adjustable numbers — the muscle fraction, the work and power per kilogram of muscle, and the leg length’s constant — set so that a 70 kilogram body has legs of forty centimetres and a jumping muscle doing sixty joules per kilogram of body. Each varies between animals by a factor of two or more. Real legs are levers whose gearing changes during the push, so the force the ground feels is not constant and the take-off time is not exactly 2L/v. The drag figure treats every body as a sphere of water density with a drag coefficient of one, which is crude for a flea, whose shape and orientation change in flight. And the measured take-off speeds are typical published values for single species, not averages over the animals in each group.

What the argument establishes is the shape: a height independent of size where muscle has time to work, a height falling as the 2/9 power of mass where it does not, and air drag cutting in below a few milligrams. Every measured jumper sits where that shape says it should, given whether it has a spring.

Still open: how fast a biological spring can be made to release #

Catapults release energy faster than muscle can, but how fast is set by the latch and the spring’s own mass, and the fastest biological movements known are much faster than a flea’s jump. The mandibles of trap-jaw ants close in a tenth of a millisecond; a mantis shrimp’s club strikes with an acceleration of over ten thousand times gravity. How latches control release, how much energy is lost to the spring’s own inertia when it unloads so fast, and what limits the power of these systems as they get smaller are the questions of a growing field that compares them across species and builds small robots on the same principles. Whether the limits found in insects are the limits of springs in general or of the materials evolution had to hand is unresolved.

The habit worth carrying away is to separate what a process costs from how fast it must be paid. Muscle work and the work of lifting both grow with mass, so every jumper rises about the same half-metre; but a flea must spend that energy in 0.43 ms, which would take 562 times the power muscle can give, and it reaches Borelli’s height only by loading a spring for a hundred times longer than it pushes. Energy is set by mass; power is set by time, and time is set by size.