Pushing the limits of energy: the potential of pole vaulting

Sport and energy: a harmonious dance

Pole vaulting is the perfect example of how energy transforms: the kinetic energy from the run-up becomes elastic energy in the pole, which then converts to gravitational potential energy, allowing the athlete to clear incredible heights like the 6.22-metre world record.

For centuries, energy played hide and seek with physicists: it was everywhere around them, but only after a long time did they manage to discover and unravel all its meanings. One of the best-known arenas where energy and its behaviour play a fundamental role is sport. The sport that best encapsulates the physical theories we are about to analyse is pole vaulting, one of the oldest disciplines and the queen of track and field.

We will explore the importance of certain fundamental laws in the study of optimal vaulting techniques. These techniques, when executed to the limits of perfection, provide the athlete with the tools to push beyond their own limits, achieving exceptional performances.

Energy: the journey from its birth to its many forms

Energy: a word we all use, yet one that is difficult to define precisely. It is more an abstract mathematical concept than a tangible reality. In general terms, the energy a body possesses represents its ability to produce a change in its own state, manifesting in many different forms.

As the renowned physicist Richard Feynman stated: "It is important to realise that in physics today, we have no knowledge of what energy is."

The discovery of energy dates back to the 17th century, when mathematicians Isaac Newton and Gottfried Leibniz measured variations in values as they raised and lowered weights along a machine, noticing that these changed depending on the speed and height at which the weights were placed.

In the world of sport, the dominant form of energy is the one linked to bodily motion, namely mechanical energy, obtained from the sum of three other types of energy: kinetic energy, gravitational potential energy and elastic potential energy.

Kinetic energy is the form of energy a body possesses by virtue of its motion. It depends on the body's speed — the faster it goes, the more energy it has — and is also proportional to the body's mass. This is evident when you try to stop a moving body: the heavier and faster the body, the more energy is needed to stop it, as rugby players know all too well.

Gravitational potential energy is an energy linked to the position (height) of a body within Earth's gravitational field. Since our bodies are attracted to the earth by gravity, it is clear that if we are standing still with our feet on the ground, this form of energy will be zero.

Finally, elastic potential energy is the energy stored by an elastic (deformable) body when it is temporarily compressed or stretched in a reversible manner. The greater the compression, the greater the elastic energy stored. This energy is usually associated with man-made objects and depends on the material from which they are made. Each body has an elasticity index, known as the elastic constant K, which is directly proportional to the elastic energy value.

The formulas for the three forms of energy are fundamental in the study of physics and apply to countless phenomena:

• Kinetic energy: K = ½·m·v²
• Gravitational potential energy: Ug = m·g·h
• Elastic potential energy: Uel = ½·k·(Δx)²

Richard Feynman, in describing one of the fundamental principles of the universe, said: "There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law — it is exact so far as we know. The law is called the *conservation of energy**. It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes... it is simply the fact that we can calculate a number, and when we finish watching nature go through her tricks and calculate the number again, it is the same."*

What Feynman describes is the explanation of the principle of conservation of energy, one of the most important laws observed in nature. In the late 19th century, about a century after the discovery of mechanical energy, a group of scientists, including the British James Joule, experimentally demonstrated that the sum of kinetic, potential and thermal energy in a system remains constant. This principle states that the total amount of energy in an ideal system (without friction) does not change over time, but simply transforms from one form to another.

One of the most representative examples of this principle in the world of sport is pole vaulting. The Department of Sports Sciences at the University of Grenoble, with the help of young athletes rising in the international scene, managed to study every single phase of this sport, revealing all its secrets from a physics perspective.

Pole vaulting seems simple. An athlete holds a pole of about 5 metres, with which they must launch themselves over a bar placed at a certain height. The vault consists of several phases: first the run-up, in which the athlete accumulates as much kinetic energy as possible; then they plant the pole in a box and use the pole's flexibility to transform kinetic energy into elastic potential energy during the pole's compression. Finally, the athlete begins the upward motion, launching into the air: the pole releases the elastic energy, converting it into gravitational potential energy, allowing the athlete to continue their upward movement.

If all these phases are executed with absolute precision and force, the result is a great vault. As of today, the world record in pole vaulting, held by Swedish athlete Armand Duplantis, stands at 6.22 metres!

Is that all? Let's discover the other variables at play

Julien Frere, head of the Faculty of Sports Sciences at Grenoble, highlighted how the model presented above is a simplified version of reality and does not consider all the factors that influence a good vault. For example, the theory of conservation of energy is valid only in the absence of friction, a condition we cannot ignore. Although modern technology has produced materials that minimise friction with air and other surfaces (such as the aerodynamic suits worn during the run-up, or a hyper-elastic pole that flexes perfectly when it touches the ground, or specific footwear), friction cannot be completely eliminated.

Therefore, the skill of the athlete, coaches and technical staff lies in minimising the dissipation of kinetic energy during the running phase and converting it as effectively as possible into gravitational potential energy which, as we have seen, translates into higher vaults. If, however, we assume an ideal situation in which everything is executed perfectly, we obtain an equation that calculates the vault height:

K₀ = Uel → Uel = Ug → K₀ = Ug

½·m·v² = m·g·h → h = v² / (2·g) ≈ 5.10 m

In theory, this result should represent a nearly unreachable limit, based on a perfect performance. However, the result obtained is much lower than the world record held by Duplantis. Where did we go wrong?

Frere and his team noticed that the error lies in the fact that the height calculated in the gravitational potential energy formula does not start from the ground, but from the centre of mass of the vaulter! Indeed, the centre of mass of a body is the point where all its mass appears to be concentrated. So if a vaulter were 2 metres tall, their centre of mass would be on average one metre higher than the ground, thus explaining the result obtained from the equations.

Another important element is the angle between the pole and the ground at the moment of the vault: the greater the angle the athlete manages to create, the more energy they will release. This is explained by the fact that by increasing the angle, we raise our centre of mass at the moment of launch and consequently increase the height from which the flight phase begins. Studies at Grenoble have shown that this angle can be maximised both by increasing speed before planting the pole and by modifying the grip height.

In conclusion, physics plays a crucial role in improving athletic performance. Understanding fundamental principles like the conservation of energy and the concept of centre of mass can lead to innovative tactics and more effective training techniques.

FAQ

What is the world record in pole vaulting?

The world record in pole vaulting is 6.22 metres, held by Swedish athlete Armand Duplantis. This extraordinary result comes from the perfect combination of run-up speed, impeccable technique and optimal energy transfer through the pole.

How does energy transform during a pole vault?

Energy transforms in three phases: first the athlete accumulates kinetic energy during the run-up, then this converts to elastic potential energy when the pole bends, finally the pole releases this energy converting it to gravitational potential energy that propels the athlete upward.

Why is the theoretical calculated height lower than the world record?

The theoretical formula calculates height from the ground, but in reality the reference point is the athlete's centre of mass, located roughly at half their height. Additionally, the pole angle and grip height add further initial elevation.

What physics principles should a pole vaulter understand?

A vaulter should understand the conservation of mechanical energy, the concept of centre of mass and the importance of the angle between pole and ground. Optimising these factors minimizes energy losses from friction and maximizes achievable height.