The Long Jump - Physical Backgrounds of the Long Jump for Coaches and Athletes [ARTICLE]
|The Long Jump - Physical Backgrounds of the Long Jump for Coaches and Athletes|
By: Vladimir Strelnitski, Max Coleman and Bill Ferguson
Originally Published in: Techniques Magazine Provided by: USTFCCCA
In this article addressed to practicing coaches and athletes we present a summary of physical phenomena involved with the long jump. We explain the mechanics of the long jump on the basis of the horizontal and vertical components of the athlete's center of mass velocity. We believe this approach presents the processes more clearly than the approach based on the total velocity and the projection angle, which has lately been used in most of the publications on the subject.
An abundant literature concerning the physical aspects of the long jump exists, but most of these research papers are written in a language that intimidates practicing coaches and athletes. Yet, there is little doubt that the understanding of the physical background of the sport a coach teaches makes the training process more efficient. This article is a result of numerous mutually beneficial discussions of the physics and techniques of the long jump by two coaches (M.C. and B.F.) and a physicist and coach (V.S.). We tried to follow the famous Einstein's remark and present things as simply as possible but not simpler.
In section 2, we formulate five questions about the technique of the long jump requiring some knowledge of physics to be answered correctly. Section 3 summarizes the concepts of physics that help answer these questions. In Sections 4 and 5 we use these concepts to explain the key elements of the long jump technique; the concept of velocity is central for section 4 and the concept of force - for Section 5. In Section 6 we elaborate on the important issue of the jump leg flexure during the board contact phase and emphasize the gradual character of the increase of the takeoff knee angle and of the leg stiffness with the increasing mastership of the athlete (increasing takeoff speed). Section 7 summarizes our conclusions in the form of brief answers to the five questions of Section 2.
We hope that the article will be helpful to coaches and athletes working on improvements of the long jump technique and interested in knowing not only what should be done for a good jump but also why it should be done.
FIVE QUESTIONS THAT REQUIRE PHYSICS TO BE ANSWERED
(1) Why should the run up for the long jump be as fast as possible?
(2) Should the jumper intentionally flex and then straighten the knee joint of the jump leg during the foot contact with the board - in order to jump higher?
(3) What is the purpose of launching the arms and the free leg up after the contact of the jump foot with the jump board ("touchdown")?
(4) Can the jumper increase the height of the free flight (and thus the time and distance of the flight) by applying an appropriate technique during the flight (such as hitch-kick)?
(5) Why should the jumper stretch the legs forward before landing?
PHYSICAL CONCEPTS TO KNOW
To understand the basic physics of the long jump one should be acquainted, as a minimum, with (1) the notion of the center of mass (COM); (2) the three Newton's laws of classical mechanics; and (3) the "vector" nature of velocity and force.
3.1. The center of mass (COM)
Slightly simplifying the strict definition, one can define COM as the point relative to which all the weights of the body's parts are mutually balanced. In other words, if the body is suspended on a string attached at the COM, it will hang in equilibrium.
3.2. Newton's laws
First Law (The law of inertia): An object stays in motion with the same velocity unless it is acted upon by an external force.
Second Law (The force-acceleration law): A force acting upon an object produces acceleration (change of the numerical value or the direction of the velocity of the motion).
Third Law (The action-reaction law): Two interacting bodies always act upon each other with equal opposite forces.
The velocities and forces involved in the long jump are "vectors" - physical quantities that are characterized not only by their numerical value but also by their direction. Graphically, a vector quantity is represented by an arrow showing the direction, the length of the arrow giving, in some units, the numerical value of the physical quantity.
An important property of vectors is the way they add to each other: The sum of two vectors is the diagonal of a parallelogram whose sides are the added vectors. If the added vectors are perpendicular to each other (for example, one is horizontal and the other is vertical), the parallelogram is reduced to a rectangle. From the "parallelogram rule" of vector addition it follows that any vector can be "resolved" into two components acting along two arbitrary directions. For example, the velocity V of the jumper's COM can be resolved into its horizontal (Vh) and vertical (V,) components (Fig. 1, pg. 20). The choice of the horizontal and the vertical directions for the two components is handy in our case, because the length of the long jump is measured horizontally, and the force of gravity with which the jumper struggles to jump farther is vertical. Fig. 2a (pg. 20) gives another example - the horizontal and vertical components of the force F exerted by the jump foot on the takeoff board during the contact phase, while Fig. 2b (pg. 20) shows the reaction force of the board R (Newton's 3rd law!) and its horizontal Rh and vertical 11, components.
The combination of the two components of a vector is totally equivalent to the parent vector. For example, the motion of the COM at takeoff will be the same if instead of considering its moving with the velocity V we consider it to participate in two independent motions - horizontally, with velocity Vh and vertically, with velocity Vv. Similarly, the mechanical effect of the force R in Fig. 2b would be the same if, instead of this force, two independent forces, Rh and R, were applied simultaneously.
In the following sections, we will show how these physical concepts help us understand the techniques of the long jump.
THE LENGTH OF THE JUMP
The long jump is a Track and Field event in which the participants are supposed to demonstrate how far they can leap forward in the horizontal direction taking off before a "foul line" drawn on the ground perpendicular to the direction of the jump. The "legal" length of the jump is the sum of three horizontal distances: (1) the distance ID, between the foul line and the COM of the jumper at the instant of takeoff (in a correct jump, the COM at this instant is always in front of the takeoff line), (2) the distance D/ between the heels of the jumper and the COM at the instant of landing (in a correct jump the heels land in front of the COM and leave the most distant mark in the sand), and (3) the distance Df that the COM covers during the free flight. The first two distances are "subsidiary." The jumpers, of course, should try to maximize them, but in the jumps of the elite athletes these distances only make 10-15% of the total legal distance, the major part being acquired in the free flight.
The motion of the COM in the free flight is described by the ballistic equation. The original form of this equation, which we will be using here, gives the horizontal length Df of the parabolic trajectory of the COM as a function of its horizontal Vh and vertical V, speeds at the takeoff and the difference h of its height at the moment of takeoff and the moment of landing (h is considered to be positive when the COM is lower at the landing than at the takeoff, as it should be in a good jump):
where g 9.8 m/s2 is the acceleration of gravity. The equation looks cumbersome because of the term within the braces, which, actually, is much less important than the pre-brace term.
The pre-brace term shows that the free flight length Df depends on the product of the horizontal and vertical speeds at the moment of takeoff. It would be great if there were a technique that would allow the jumpers to attain simultaneously their personal records in Vh and V„. Unfortunately, such a technique is unknown. In any real technique, an increase in the vertical speed is only possible at the expense of a decrease in horizontal speed, and therefore the jumper has to look for a compromise that would not necessarily maximize each of these velocity components but would maximize their product.
The major characteristic feature of the running long jump is a drastic difference between the maximum possible vertical speed and maximum possible horizontal speed at takeoff: the latter is more than twice as great as the former. Let us see why.
The horizontal speed is limited by the highest possible speed of the run-up sprint, which is currently around 11 m/s. Because of the unavoidable loss of the horizontal speed due to the braking inter-action of the jump leg with the ground between touchdown and takeoff (see below), one can admit 10 m/s as a realistic upper limit for the horizontal takeoff speed.
The upper limit for the vertical takeoff speed can be estimated by the maximum height H to which the jumper can lift his or her COM after takeoff. The equation describing the connection between V, and H is:
The elite male high jumpers are capable of lifting their COM by about 1.0 m from its takeoff height, which, according to the above equation, corresponds to the initial vertical speed of approximately 4.4 m/s. Such an extreme vertical speed is obviously unachievable for a long jumper because of technical limitations; one can take 4.0 m/s as a "very optimistic" upper limit.
With these upper limits for Vh and V, and with the COM takeoff to landing drop of 0.6 m (achievable by the best jumpers), equation (1) gives the maximum theoretical range of the free flight part of the jump: Dr9.5 m. The horizontal shifts between the COM and the feet at the moments of takeoff and landing (ID, and D1) can add together up to ,..1m to the legal distance D, which results in Ds10.5 m. This can be compared with the current long jump world record of 8.95 m.
The stricter limitation for the vertical takeoff speed than for the horizontal speed allows for understanding of why the projection angle (Fig. 1) in the high-class long jumps is much smaller than the well known optimal angle for a simple projectile, the famous "45°."The projection angle a is determined by the ratio of the vertical and horizontal component of the velocity (Fig. 1). In trigonometry, this ratio is called the tangent (tan) of the angle. So, a is the angle whose tangent equals Vv/Vh. Using the maximum possible values for the speeds, V, . 4 m/s and Vh .... 10 m/s, we find: tan a = V, /Vh ,..0.40, and any modern calculator (e.g. the one in your iPhone) will tell you that the angle having this value of tangent is a ,..22°. And, indeed, the average projection angle for elite long jumpers, both men and women, is a .21° (Linthorne 2007).
The less important, braced term in equation (1) describes the correction to Df due to the lowering of the COM at the instant of landing relative to its height at the instant of takeoff. A 1.8 m tall male jumper can drop his COM by h ,..0.6 m. Equation (1) shows then that for the 8-m range jumps (Vv.3.5 mis , see below) this lowering of the COM increases the distance of the flight Df by . 20%. It is a considerable improvement, and one of the reasons why the jumper should stretch the legs forward before landing (the second reason being, obviously, the increase of the distance Dd. It is interesting that for a less experienced athlete, for whom the vertical takeoff speed is lower, the same drop of the COM at the landing would increase the distance of the flight more efficiently. For example, for the 6 m range jumps (Vd.2 m/s ) the increase of the distance due to the same (0.6 m) drop of the COM is already.50%. Still, the jumpers must work on increasing both their horizontal and their vertical takeoff speed, because the pre-brace term in equation (1) is more important in creating the length of the jump than the braced term.
Note that it is the COM of the body what moves along the ballistic parabola in free flight. The position of the COM is not fixed in the body, it changes when the relative positions of the body's parts change. Thus, what moves exactly along the parabola is not a fixed part of the body (like the pelvis, for example) but a mathematical point migrating in (or out) of the body in accordance with the momentary relative positions of the body's parts. The position of the COM at each instant can be calculated if the masses and spatial positions of all the parts of the body are known. We emphasize that after the takeoff and up to the landing no relative movements of the parts of the body can change the free flight trajectory of the COM. In particular, no leg or arm technique during the free flight can make the COM fly higher, and thus increase the length of the jump. All the leg, arm, torso, and head movements in the flight are only aimed at creating the optimum body position for landing.
THE CREATION OF THE VERTICAL SPEED
It is important to understand that the vertical speed in the long jump is due exclusively to the reaction force developed in the takeoff board in response to the pressure from the jump foot. More precisely, the upward acceleration is due to the vertical component of this reaction force (Fig. 2b).
What should the jumper do to create and maximize this reaction force?
An obvious answer seems to be: To use the "spring" of the jump leg by intentionally bending and then powerfully extending it to produce a strong pressure on the board and get the correspondingly strong reaction response. Don't we do just this (with both legs) in the standing long jump?
Well, in the high-class running long jump, there is simply not enough time for that! The time of the jumper's contact with the board can be estimated as the time the COM moves from its position (behind the jump foot) at touchdown to its position (in front of the jump foot) at takeoff - a distance of about lm. The horizontal speed of the motion is .10 m/s, and thus the time of the contact is only on the order of 1/10 of a second! The "spring" of the leg that would first deeply bend and then fully extend would be too "soft" to accomplish its work during such a short time.
Here is a simple example that may help understand this. Imagine a ball falling vertically on a horizontal platform that is available only for a small fraction of a second after the contact with the ball, after which it is swiftly removed downward. If the ball is soft, it will not have enough time even to compress in full after the moment of contact. Before it fully compresses and bounces off, the platform is already gone, and the ball will keep moving downward, instead of rebounding up or, at best, it will rebound with a small upward speed. In order for the ball to rebound with full efficiency in the available short time of contact, it must be stiff enough, which means that a small deformation creates a strong reaction force in its material. It is clear, intuitively, that the smaller the deformation necessary for the full rebound, the shorter the needed time of contact. [An interested reader can find the details of the bouncing ball physics in Cross (1998), where it is shown, in particular, that the time of the ball contact with the surface decreases with the increase of the ball's stiffness.]
This brings us to the technical solution that is used by experienced long jumpers. They do not bend their jump leg after the touchdown, at least they do not bend it intentionally. On the contrary, they do their best to resist the increase of the knee flexion at the touch down, making the leg as stiff as possible on the last stride before the touchdown. The knee joint will bend a little after the touchdown, because of the very strong shock, but the jumpers try to minimize the flexure. As a result, the intentionally stiffed leg behaves like a very stiff spring. This spring absorbs the board's reaction force quickly and quickly transmits it to the whole body.
The movement of the COM upwards starts immediately after the touchdown. The COM lifts even while the knee of the jump leg is still (involuntarily) flexing. The lifting is due to the pivoting of the body around the jump foot ("stopped" by the board) and by throwing the arms and the free leg up. Throwing these parts of the body up increases the vertical pressure of the jump foot on the board. This effect of "kick back" is the consequence of the second and third Newton's laws and can be easily verified with an ordinary bathroom scale. Stand on the scale with your two feet and notice your weight. Then throw your arms and/or one leg up: you will see the scale showing a considerable instantaneous "increase of weight" (increase of the pressure on the scale) and then a quick return back to the normal value. The increase of the jump foot pressure on the board produces an additional upward reaction force from the board. After the moment of maximum knee flexion, the forcefully extended muscles of the jump leg thigh react by a fast "concentric contraction" which extends the jump leg increasing the pressure on the board (and thus the vertical reaction force of the board) even more. Finally, a small addition to the pressure on the board (and thus to the upward reaction force) is produced by the extension of the ankle joint before the takeoff. In the high-class jumps, the combination of these actions imparts to the COM an upward velocity as high as . 3.5 m/s by the moment of takeoff.
Contrary to the vertical component of the board reaction force, the horizontal component (Fig. 2b) plays a negative
role - it brakes the horizontal speed up to the instant the COM of the body pivoting around the jump foot has moved forward to the position just above the foot. After
this moment, the jump foot presses the board backward, and the opposite reaction force gives the athlete some impulse forward, compensating a little for the loss of the horizontal speed in the first phase. This compensation is not complete: typically, by the moment of takeoff, 15-20% of the touchdown horizontal speed has been lost. Thus, the touchdown-to-takeoff phase allows the jumper to create the maximum possible upward speed at the expense of a 15-20% loss in the horizontal speed.
THE TECHNICAL GROWTH
The above discussion concerned the physical backgrounds of the high-class long jumps. Here we focus on some tendencies in the technical growth of non-elite athletes.
Two typical differences of a beginning long jumper from an elite jumper are: (1) a lower run-up speed and (2) weaker muscles and joints of the legs.
A lower run-up speed (Vru) entails a longer foot contact time (takeoff duration). An experiment, in which an elite athlete was asked to jump full force but with various lengths (and thus various speeds) of the approach, demonstrated a decrease of the takeoff duration from .0.20 s to .0.12 s when the run-up speed increased from ,=5 m/s to .11 m/s (Linthorne 2007). The experiment showed that the athlete tried to jump as high as possible, regardless the speed of approach - to achieve the maximum length of the jump by increasing the time of the free flight. So, the vertical take-off speed remained almost constant, close to the maximum for this athlete, whereas the horizontal takeoff speed grew almost proportionally to the run-up speed. As a result, the projection angle of the jump (determined by the ratio of the vertical and horizontal takeoff speeds) was gradually decreasing, from .40° at Vru ,..5 m/s to ,..20° at Vru .10 m/s.
Another important result from this experiment was the observed decrease of the touchdown knee angle with the decreasing run-up speed - from .170° at Vru . 11 m/s to .150° at Vru . 5 m/s. As discussed above, the touchdown shock in the high-class (fast run-up) long jumps produces unintentional flexure of the knee joint. But since the intensity of the shock decreases with the decreasing run-up speed, the observed increase of the knee flexion could only be intentional. By flexing and powerfully straightening the leg, the jumper increases the foot pressure on the jump board and achieves the maximum possible upward speed, compensating for the lower efficiency of the "pivoting" acceleration mechanism at lower runup speeds. Using the muscles of the jump leg intentionally for an additional push on the jump board becomes possible because of the longer takeoff duration when the run-up speed is lower.
Supplementing these results with the standing long jump, where the "pivoting" mechanism of upward acceleration is totally absent, and both the vertical and the horizontal accelerations are obtained only by an intentional flexing/straightening of the legs, one can come to the conclusion that from the standing long jump through the running jumps with growing run-up speeds, to the highest run-up speed jumps, the role of the intentionally used muscles of the jump leg to create the maximum upward velocity gradually decreases, whereas the role of the "pivoting" mechanism on a pre-activated leg gradually increases. Therefore, when working with inexperienced long jumpers the coach should make efforts to gradually attain: (1) an increase of the approach speed, (2) an increase of the leg muscle strength, and (3) a decrease of the intentional knee flexure at the touchdown. The last two requirements aim at making the "spring" of the jump leg as stiff as possible for high-class jumps.
We summarize our conclusions by briefly answering the five questions formulated in Section 1:
(1) Although the relation between the horizontal and vertical takeoff velocity components is complex, the experience shows that an increase in the run-up speed increases the product of the two velocity components, to which the distance of the jump is proportional [equation (1)].
(2) Flexing and straightening the knee of the jump leg intentionally softens the "spring" of the leg. This increases the time of the leg re-bouncing beyond the very short time of the foot contact with the board available in a high-class (high runup speed) jump. At the touchdown, the jump leg must be as stiff as possible and its foot must be put on the board flat to avoid a premature ankle joint flexion.
(3) Launching the arm(s) and the free leg upward upon the touchdown has a triple effect: it increases the pressure of the jump foot on the board (and thus the upward reaction of the board); it raises the takeoff height of the COM (and thus the length of the free flight); and it helps to keep the torso vertical by opposing the forward rotation caused by the stopping of the jump foot by the ground at the touchdown.
(4) After the takeoff, no relative movements of the body parts can alter the motion of the COM along the ballistic trajectory whose parameters (including the maximum height of the flight) are predetermined by the horizontal and vertical velocities of the COM at the instant of takeoff.
(5) Before landing, the jumper stretches the legs forward in order to (i) maximally lower the COM at the moment of landing and (ii) maximally increase the horizontal distance between the feet and the COM at landing - both effects increasing the distance of the jump.
Cross, R. C. (1999, The bounce of a ball, Am. J. Phys. 67, 222-227.
Linthorne, N.P. (2007). Biomechanics of the long jump. In Routledge Handbook of Biomechanics and Human Movement Science, Y. Hong and R. Bartlett (Editors), Routledge, London. pp. 340-353.
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Vladimir Strelnitski has PhD in astrophysics. Director Emeritus of the Maria Mitchell Observatory (retired in 2013). He taught the Physics of Movement at Springfield College in Massachusetts and currently coaches the hammer throw at Springfield.
Max Coleman is the sprints, jumps and hurdles coach at NCAA Division II Catawba College in Salisbury North Carolina.
William Ferguson is Currently Assistant Coach for Track and Field / Cross Country at Moravian College.