## Introduction

As all formulas for intensity factors and most of the metrics require a normalized power, 3record tries to estimate the power even for activities that do not have the power information.

This is based on speed and slope according to literature formulas explained below. Note that these are only approximations though and can be quite far from the actual power produced depending on the circumstances.

## Power estimation

### Methodology

Power is defined as the product of force times velocity

$P = F\cdot v$ and therefore a good way to calculate the power is to estimate the forces one had to overcome to reach a certain speed. These forces are detailed in the paragraphs below according to the specificity of the sport, but fall generally in the following categories:
Drag (also referred as aerodynamism)
Drag force is the resistance that air or water opposes to your movement. It depends on the square of velocity ($F\propto v^2$) and on the density of the medium. The square dependence makes it much stronger and noticeable in biking than in running. Similarly, high density of water and slow swimming speed makes it the dominant contributino in swimming.
Gravity
Gravity force $F=mg$ is what makes it difficult to overcome a climb. It depends on the slope of the climb $s$.
Resistance and friction
Friction generally depends linearly on speed but can have quite complicated forms depending on the source of losses. Sometimes, this is included globally in an efficiency factor to account for all the force not converted into useful power to produce speed (some being lost in heat for instance).
Kinetic energy
Speed changes (changes in kinetic energy by $E=\frac{1}{2}mv^2$) require the application of a force $F=ma$ and therefore translate into power.

### Swimming

In swimming, the main resistance against movement is due to the drag of water. We can use for instance the formula from Toussaint available in [1, Skiba SwimScore] and used by Skiba for his SwimScore (note that the NP is still calculated with the 4th power and constant-weighted averages in 3record and not the 3rd of exponentially-weighted averages as in the SwimScore).

The force depends then on the square of the velocity $v$ and, consequently, the power is a factor times the third power of velocity.

\begin{aligned} P &= \eta^{-1} C_{drag}\cdot v^3\\ &= \eta^{-1} \left(2+0.35m\right) \cdot v^3 \end{aligned} with $\eta=0.6$ an efficiency factor and $m$ the athlete's mass.

### Biking

It is quite difficult in biking to make accurate estimations of power. Martin et al. have published a model [2, Martin Cycling power] that depends on speed $v$, slope $s$ and wind speed. Wind speed as well as drafting are neglected as the relevant factors cannot be known. The results are therefore only rough estimations.

The power is split into different contributions:

$P = \eta^{-1} \left(C_{aero} + C_{roll} + C_{WB} + C_{slope} + C_{kin}\right) \cdot v$ where $\eta$ is an efficiency factor assumed to 1 and the following factors:

• #### Aerodynamism

\begin{aligned} C_{aero} &= \frac{1}{2}\cdot \rho \cdot \left( C_d A_f + F_w\right) \cdot \left(v+v_{wind,front}\right)^2\\ &= \frac{1}{2}\cdot 1.2 \cdot 0.26 \cdot v^2 \end{aligned} with $\rho$ the air density, $C_d$ the coefficient of aerodynamism, $A_f$ the frontal area and $F_w$ the added equivalent frontal area from the spokes.
• #### Rolling resistance

\begin{aligned} C_{roll} &= \cos\tan^{-1}(s)\cdot C_{rr} \cdot m_{tot} \cdot g\\ &= \cos\tan^{-1}(s)\cdot 0.0032 \cdot m_{tot} \cdot 9.81 \end{aligned} with $C_{rr}$ the rolling resistance, $m_{tot} = m_{rider} + m_{bike}$ the total mass and $g$ the gravity constant.
• #### Wheel bearing friction

$C_{WB} = \left(91+8.7\cdot v\right)/1000$
• #### Slope/Potential energy

\begin{aligned} C_{slope} &= \sin\tan^{-1}(s) \cdot m_{tot} \cdot g\\ &= \sin\tan^{-1}(s) \cdot m_{tot} \cdot 9.81 \end{aligned}
• #### Kinetic energy

\begin{aligned} C_{kin} &= \frac{1}{2} \cdot \left(m_{tot}+I/r^2\right) \cdot \left(v^2 - v_0^2\right) / d\\ &= \frac{1}{2} \cdot \left(m_{tot}+0.14/0.311^2\right) \cdot \left(v^2 - v_0^2\right) / d \end{aligned} with $I$ the inertia from the wheels and $r$ their radius.

$P = \left(0.156v^2+0.031392\cdot m_{tot}\cos\tan^{-1}(s)+(0.091+0.0087v)+9.81m_{tot}\sin\tan^{-1}(s)\right)\cdot v$ which can be approximated by $P \approx 0.03v\cdot\left(5.2v^2+(1+327s)m_{tot}\right)$

Note that you can find a similar online calculator at http://www.analyticcycling.com/

### Running

In running, more formulas have been develop to quantify the effort of running at a certain speed $v$ on a certain slope $s$.

One example is the one used by Skiba for his GOVSS metric [3, Skiba GOVSS], yielding the following formula:

$P = \left(C_{aero} + C_{slope}\cdot\eta\cdot m + C_{kin}\right) \cdot v$ where $\eta=(0.25+0.054v)\cdot(1-0.5v/8.33)$ is an efficiency factor and $m$ is the runner weight, and the following factors:

• #### Aerodynamism

\begin{aligned} C_{aero} &= \frac{1}{2}\cdot \rho \cdot \left( C_d A_f \right) \cdot v^2\\ &= \frac{1}{2}\cdot 1.2 \cdot 0.9 \cdot A_f \cdot v^2 \end{aligned} with $\rho$ the air density, $C_d$ the coefficient of aerodynamism, and $A_f=0.2025\cdot 0.266 \cdot h^{0.725} \cdot m^{0.425}$ the frontal area derived from athlete's height and mass.
• #### Slope/Potential energy

\begin{aligned} C_{slope} &=& 155.4s^5-30.4s^4-43.3s^3+46.3s^2+19.5s+3.6 \end{aligned} a fitting formula from [4, Minetti Running slope].
• #### Kinetic energy

\begin{aligned} C_{kin} &= \frac{1}{2} \cdot \left(v^2 - v_0^2\right) / d\\ &= \frac{1}{2} \cdot \left(v^2 - v_0^2\right) / d \end{aligned}

Leading to the final formula which can be approximated for small slopes and speeds by

$P \approx (0.9+5s)\cdot m\cdot v^2$

## References

References in text

[2, Martin Cycling power]
Validation of a Mathematical Model for Road Cycling Power, J. Martin et al., J Appl Physiol 1998, http://www.naspspa.org/AcuCustom/Sitename/Documents/DocumentItem/2415.pdf.
[3, Skiba GOVSS]
GOVSS, P. Skiba, http://www.physfarm.com/govss.pdf.
[4, Minetti Running slope]
Energy cost of walking and running at extreme uphill and downhill slopes, Minetti et al, J Appl Physiol 2002, http://jap.physiology.org/content/jap/93/3/1039.full.pdf.