Energy in the 100 m Sprint

Energy is usually defined as the capability for doing work. For example, when Usian Bolt set the world record of 9.58 s in the 100 m sprint (Berlin, 2009), he used energy to accelerates his body mass of 86 kg (190 lb) to his top speed of around 12 m/s (around 27 mph) at around the 65 m mark. This is about the fastest running speed, achieved by Bolt, Maurice Greene and Donovan Bailey (1996). Record marathon times are around 2 hours, 5 minutes (about 12.77 mph), and mile times around 4 minutes (15 mph).

Figure \(\PageIndex{1}\) Jamaican world record holder Usain Bolt

The energies involved are surprising when we look at them in detail below.

Kinetic Energy

Kinetic energy is energy due to motion, and is represented by E_k. For the bird moving in a straight line, the kinetic energy is one-half the product of the mass and the square of the speed:

\[E_k = \dfrac{1}{2} mu^2 \label{1}\]

where

• \(m\) is the mass of the object in kg
• \(u\) is the speed of object in m/s

Example \(\PageIndex{1}\): Kinetic Energy of Usain Bolt

Calculate the kinetic energy of Usain Bolt at his maximum speed of 27 mph (12 m s^–1) if he weighs 86 kg.

Solution

\(\large E_{k} = \dfrac{1}{2} m u^{2} = \dfrac{1}{2} \times 86 \text{ kg} \times ( 12 \text{ m} \text{ s}^{-1} )^{2} = 6192 \text{ kg}\text{ m}^{2} \text{ s}^{-2}\)

The collection of units kg m² s^–2 is given the name Joule in the SI system after James Joule (see below). In other words the units for energy are derived from the SI base units kilogram for mass, meter for length, and second for time. A quantity of heat or any other form of energy may be expressed in kilogram meter squared per second squared.

Calories

In everyday life, we often measure energy in Calories, and relate them to food energies. The calorie used to be defined as the energy needed to raise the temperature of one gram of water from 14.5°C to 15.5°C but now it is defined as exactly 4.184 J. The capital "C" in food Calorie indicates that this is really kilocalories (1000 calories = 1 Calorie). We know that food calories heat our bodies and allow us to do useful work (and maybe gain weight), and we'll see how they're measured, and consumed, in the next sections.

The energy in Calories due just to Bolt's velocity is \(\text{6192 J} \times \dfrac{\text{1 cal}}{\text{4.184 J}} \times \dfrac{\text{ 1 Cal}}{\text{1000 cal}} = \text{1.5 Cal}\)

So an 86 kg body has very little energy at 27 mph. Why does it seem to use so much energy to run that fast? The physiological...and physics...details have been studied extensively . ^[1]

The actual amount of energy spent by Bolt in a 100 m race is about 116 117 J (or about 277 Cal). ^[2]. This includes a miniscule amount to overcome wind resistance (possibly as low as 0.020 Cal for the race)^[3], but mostly energy required to move legs and arms back and forth and other body movements and functions. Finally, potential energy is involved.

Potential Energy

Potential Energy is energy that is stored in a body by rising in height (in the case of Bolt when he takes a step), or by other means. It frequently comes from separating things that attract, like the body of a Bolt and the Earth that attracts him, or by pulling magnets apart, or pulling an electrostatically charged balloon from an oppositely charged object to which it has clung.

About 0.048 Cal/s (200 Watts) was probably spent on the up and down motion of Bolt's body.

Potential Energy is abbreviated E_P and gravitational potential energy is calculated as follows:

\[E_P = mgh \label{2}\]

Where

• \(m\) is the mass of the object in kg
• \(g\) is the gravitational constant, \(9.8\; m\, s^2\)
• \(h\) is the height in \(m\)

Notice that E_P has the same units, kg m² s^–2 or Joule as kinetic energy.

Example \(\PageIndex{1}\): Potential Energy of Usain Bolt

How much potential energy is stored in Bolt's body if he raises his center of mass 2.0 cm in each step?

Solution:

\(\large E_{P} = mgh = 86 \text{kg} \times 9.8 \text{m} \text {s}^{-2} \times 0.020 \text{m} = 17 \text{kg m}^2 \text{s}^{-2} \)

This is 0.0041 Cal/step, of if the race is 100 steps, a total of .41 Cal. This energy could come at the expense of kinetic energy, requiring the runner to slow down, or it could be supplied by metabolic processes.

When we try to understand the energetics of a race in terms of body energy, our reasoning depends on The law of conservation of energy, which states that energy cannot be created or destroyed under the usual conditions of everyday life. Whenever there appears to be an increase in energy somewhere, like raising the center of mass of the body, there is a corresponding decrease somewhere else, like the body's kinetic energy or in chemical energy from food. There are clearly many forms of energy.

The first careful experiments to determine how much work was equivalent to a given quantity of heat were done by the English physicist James Joule (1818 to 1889) in the 1840s. In an experiment, Joule connected falling weights through a pulley system to a paddle wheel immersed in an insulated container of water. The moving paddles transferred the energy of the falling weight into turbulent heat in the water. This allowed Joule to compare the heat energy change of the water to the E_P of the weights, and understand how potential was related to heat energy.

From ChemPRIME: 3.6: Energy

References

1. Wagner, G. "The 100-meter dash: Theory and experiment". Physics Teacher, 36 (3), 1998, p. 144
2. O. Helene, M. T. Yamashita, Am. J. Phys. 78, 307 (2010); http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1952v2.pdf
3. Margaria, R. European Journal of Applied Physiology and Occupational PhysiologyVolume 25, Number 4 / December, 1968, p 352-360; www.springerlink.com/content/mr85778772370374/
4. O. Helene, M. T. Yamashita, Am. J. Phys. 78, 307 (2010); http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1952v2.pdf

Contributors and Attributions

• Ed Vitz (Kutztown University), John W. Moore (UW-Madison), Justin Shorb (Hope College), Xavier Prat-Resina (University of Minnesota Rochester), Tim Wendorff, and Adam Hahn.