The jump that lands as softly from any distance

A ski jumper on a large hill leaves the take-off at about 25 metres a second and lands well over a hundred metres away, having dropped more than fifty metres in height. A drop of fifty metres onto flat ground would be fatal. A drop of three or four would injure most people. Yet the landing on a ski-jumping hill is routine, and a good jumper lands with a slight bend of the knees and skis away. Nothing in the jumper absorbs fifty metres of fall. The hill does it, by being the right shape.

What hurts on landing is not speed but the speed into the surface — the part of the velocity perpendicular to the ground at the point of contact. That component has to be stopped in the short distance the legs can bend, while the component along the surface simply carries on as sliding. A landing slope that runs nearly parallel to the flight path leaves very little speed into it, however fast the jumper is going. The useful measure is the equivalent fall height: the height from which a person would have to be dropped onto flat ground to hit it as hard,

$$h_{\text{eq}} = \frac{v_\perp^2}{2g},$$

where $v_\perp$ is the speed into the surface. A jumper landing at 28 m/s on a slope that meets the path at six degrees has $v_\perp = 28 \sin 6^\circ \approx 2.9$ m/s, an equivalent fall of about 0.4 m: a hop off a low step.

The slope that is the same for everyone #

The difficulty is that jumpers do not all fly the same path. The take-off is fixed, but speeds differ — with conditions, with skill, with the snow — and each speed gives a different parabola landing somewhere else. A straight slope is parallel to the path of only one of them. The question is whether a single surface can meet every path from the take-off at the same equivalent fall height.

It can, and the surface is found by integration. Pick a point below the take-off. Exactly one parabola from the lip at the take-off angle passes through it; its launch speed $v_0$ is found by solving the trajectory equation for that point, and with $v_0$ known, the jumper’s velocity there — speed $V$ and direction $\alpha$ — follows. For the surface to give an equivalent fall height $h$ at that point, its slope angle $\varphi$ must satisfy $V\sin(\varphi - \alpha) = \sqrt{2gh}$, so

$$\frac{dy}{dx} = \tan\!\left(\alpha + \arcsin\frac{\sqrt{2gh}}{V}\right).$$

That is an ordinary differential equation for the surface. Start it at a point near the lip and step it forward, and it draws the landing hill.

The surface in the picture was not drawn by eye. At every step it took the slope that the path through that point requires, and the five marked landings, from 8 to 16 m/s, each meet it with the speed of a half-metre drop. Its shape is the shape that the family of parabolas asks for: steep close to the lip, where slow jumpers come down steeply, and bending gradually to follow the faster, flatter paths that land further out. The surface is not one of the parabolas, and it is nothing like the envelope that everywhere a throw can reach found bounding every path of one speed thrown at every angle. Here the angle is fixed and the speed varies, the paths are nested one inside the next, and the surface cuts each of them at a small, carefully chosen angle.

Ski-jumping hills have been shaped to follow the flight since the early twentieth century, by experience and by profiles fitted to measured flights. The explicit equation for a surface with constant equivalent fall height was worked out much more recently by engineers studying injuries in snow parks, notably James McNeil and Mont Hubbard, whose analyses of recreational jumps showed that the surface can be computed directly and that many jumps then being built were far from it.

Three landings for one kicker #

The difference a shape makes is clearest for the jumps where it is most often wrong: the kickers of a snow park, which throw riders upward rather than down.

The table-top is the common design: a kicker, a flat deck at the same height, and a sloping landing beyond it. A rider going fast enough clears the deck and lands on the slope with a modest impact. A rider going slightly too slowly lands on the flat deck from the top of the arc, and on flat ground the whole vertical speed is speed into the surface. The worst case is just short of the edge, where the flight is longest and the deck still flat: an equivalent fall of 1.7 m here, and several metres on the larger kickers built for experienced riders. The table-top is safest for those who need it least and most punishing for the rider who misjudges by a little. Riders going much too fast overshoot onto the slope and meet it more steeply than it falls, which is a second failure from the same cause.

A straight slope from the lip has the opposite problem. Slow riders come down close to the lip at a steep angle to a slope that falls away less steeply than they do, and their equivalent fall rises quickly as speed drops. The designed surface stays at half a metre across the whole range of speeds it was built for, which is the point. It is not more lenient on average; it is equally lenient everywhere.

What gentleness costs #

There is a price, and it is height. A softer landing requires a surface that follows the paths more closely, and the paths go down steeply.

Halving the equivalent fall height from a metre to half a metre more than doubles the vertical drop the hill needs over the same horizontal distance. This is why large ski-jumping hills are on large hillsides. A landing hill for jumps of a hundred metres has to fall away as fast as the jumpers do, and a site without that height cannot have a long, gentle landing; it can have a short one, or a harder one. It also explains why the landing hill of a ski jump is the most expensive part of the structure. The take-off and in-run can be built on a tower. The landing has to be the ground itself, shaped over a hundred metres to a profile the flight dictates.

The relation runs through the formula for the surface. The slope departs from the path by an angle whose sine is $\sqrt{2gh}/V$. For a small $h$ and a fast jumper the departure is a few degrees, so the surface is nearly the path itself, and the path falls at whatever rate gravity gives it. Gentle landings are expensive because a parabola falls quickly.

Up, down and the knoll #

The take-off angle changes the shape of the landing more than its size.

Ski-jumping take-offs point slightly downward for exactly this reason: a jumper leaving a downward ramp is already following the hill, the landing can begin directly below, and the paths of different jumpers stay close together. A snow-park kicker throws riders upward to give them time in the air for tricks, and the price is a landing that has to rise to meet the slow ones and then fall to follow the fast. The knoll of a well-built jump is that rise. Leave the fall out and keep the knoll flat for its whole length, and the result is the table-top deck.

Distance is still sensitive #

A constant equivalent fall height does not make the jump predictable.

The landing distance depends on the square of the speed in a flat-ground throw, as the angle that throws furthest found, and the sloping ground stretches that dependence further because a faster jumper falls further before meeting it. A metre per second of difference at the take-off — a slightly slower snow, a slightly worse crouch — moves the landing by several metres. That sensitivity is why competition ski jumping adjusts the in-run start gate between rounds, and why the margin of a landing hill at its far end, where the slope flattens into the outrun, is where the serious accidents happen: a jumper who flies beyond the designed section lands where the slope has already eased, with a large speed into it.

What a pair of legs can stop #

Half a metre is not an arbitrary design value. It is roughly the height a fit adult reaches in a standing vertical jump, and that is no coincidence. The jump that does not get higher with size found that the height of a jump is set by the work muscles can do over the distance the legs straighten, which scales the same way for a flea and a person. Landing is the same calculation run backwards: the legs bend through about the same distance, a few tenths of a metre, and the muscles absorb about the same work. A body can stop, comfortably and repeatedly, roughly the speed it can produce. So an equivalent fall height of half a metre asks a jumper to land as if from one of their own jumps.

That gives a natural scale for everything else. A one-metre equivalent fall asks the legs to absorb twice the energy over the same bend, which a trained athlete does without harm and an untrained one feels. Two metres is a hard landing for anyone. Beyond that, the legs cannot stop the body within their travel and the rest of the deceleration is taken by the hips, the spine or the ground, which is where injuries come from. The table-top’s 1.7 m sits in the range where a skilled rider is fine and a beginner is not, which is the observed pattern of injuries in snow parks.

Aircraft are landed on the same principle, with the path shaped instead of the ground. An airliner comes down a three-degree glide slope at about seventy metres a second, which would mean a sink rate of nearly four metres a second, an equivalent fall of most of a metre, if it flew straight into the runway. In the last few seconds the pilot raises the nose and flares, bending the path until it is nearly parallel to the runway, and a normal touchdown is made at a sink rate of a metre a second or less: a drop of a few centimetres, at a speed of seventy metres a second. A carrier landing, where there is no room to flare, accepts sink rates several times higher, and the undercarriage and airframe are built to take what is in effect a drop of a metre or more on every landing. The choice is the same as on the hill: match the path to the surface, or pay for the mismatch in structure.

Landings, tangents and a pile of chain #

The idea that a landing should meet the path at a small angle is a tangency argument in disguise. The best throw is a tangency showed that the furthest throw onto any slope is the one whose path just touches the envelope of all paths, so that a small change of angle changes nothing at first order. The landing surface is the opposite construction: instead of the path touching a fixed curve, the curve is made to nearly touch every path, each at the same small angle. A surface with zero impact would have to run along the path through each of its points, and a curve that does that is simply one of the paths: matched perfectly to one speed and to no other. The half-metre is the price of a surface that crosses the whole family.

The construction is also a cousin of one curve answers every slope, where a single curve answered the question of the best throw onto every incline at once: here a single surface answers the question of the gentlest landing for every speed at once, and both exist because the family of paths has one free parameter. The quantity being controlled is momentum perpendicular to a surface. The pile that lands heavier than it weighs found the same quantity deciding the force of a falling chain on a scale: what the scale feels is the weight already landed plus the rate at which momentum arrives, and the second term depends only on the speed into the pan. A body landing on a slope that matches its path is a chain falling onto a scale tilted to receive it, with most of the momentum carried on along the surface instead of stopped by it.

And the energy is not lost. The jumper who drops fifty metres still has the speed of a fifty-metre fall when they land, but almost all of it is along the hill. The hill that gives it back traced how a frictionless surface trades height for speed and back without loss; the landing hill hands that speed on to the outrun, where it is dissipated slowly by a long flat run and a gentle uphill, rather than in the tenth of a second the knees have.

What the picture cannot show #

The jumper here is a point with no air. A real ski jumper is an aerofoil, far more so than the thrown ball whose drag the angle that drag moves followed: in flight the body and skis generate lift comparable to a sizeable fraction of their weight, which flattens the path and lengthens the jump, and drag that slows it. Ski-jumping hills are designed from flight paths that include those forces, measured and modelled, and their profiles differ from the vacuum surfaces drawn here in detail, though not in principle. A snow-park rider travels much more slowly and the point-mass model is closer to the truth there, which is why the figures use snow-park speeds for most of the comparisons.

The equivalent fall height is also a simplification of an injury mechanism. What injures is the deceleration and how it is distributed through the body, which depends on how the legs flex, whether the landing is on the feet, and the state of the snow. A soft snow surface spreads the impact over more distance and reduces the peak force for the same speed into it. The measure ranks landings fairly; it does not predict injuries. It also ignores the component of speed along the surface, which matters if the surface curves sharply upward, as at the transition into a flat outrun, where the jumper is pressed into the snow by the curvature even with no impact.

The designed surface works only for the take-off it was integrated from. A rider who leaves the kicker at a different angle — by crouching late, or by a rut in the lip — follows a path from a different family, and the surface is no longer matched to it. Snow also changes: the landing surface of a real jump is shovelled and groomed, and it drifts from its designed profile within days unless maintained.

Still open: how much equivalent fall a person can take #

The engineering is clear once a design value is chosen: integrate the equation and build the hill. The harder question is what value to choose. Proposals for snow-park design have suggested limits around a metre and a half of equivalent fall height, on the grounds that a fall from that height onto the feet is survivable for most people, but the data relating impacts to injuries in skiing and snowboarding are sparse and confounded by technique, by skill and by the many landings that end on something other than the feet. Whether standards for terrain parks should specify a maximum equivalent fall height, and what it should be, has been argued in the engineering literature and in courts, and no consensus figure exists. The physics says that any value can be built if there is height enough; how low it should be is a question about bodies, and it has not been settled.

What a landing costs is the speed into the surface, $\sqrt{2gh_{\text{eq}}}$, and a slope integrated from the flight paths — at each point taking the angle that leaves exactly that speed — lands every jumper from one take-off as if dropped half a metre, while a flat deck lands the rider just short of its edge from the whole height of the flight, 1.7 m from a 25° kicker; softness costs height, 43 m of drop in 50 m for half a metre against 19 m for one. The habit worth carrying is to ask which component of a motion has to be stopped, because a surface that turns the rest aside can make a long fall harmless.