Welcome back to Sports & Building Aerodynamics in the week on the 100 meter sprint aerodynamics. In this module, we're going to focus on a mathematical-physical model of running. We start again with the module question. A top athlete runs the 100 meter sprint at zero wind conditions in a time below ten seconds, which is a very good performance. At which distance d, does she or he reach his or her maximum speed? Is it A) At the start, so at a distance of zero meters. Is it B) Just after the start. Is it C) At about half the distance, so 50 meters. Is it D) Just before the finish line. Or is it E) At the finish line, so at 100 meters. Please hang on to your answer and we'll come back to this question later in this module. At the end of this module you will understand the various stages of the 100 meter sprint. You will understand the background of a mathematical-physical model of the 100 meter sprint. And you will understand the time-velocity curve of the sprint. There are actually many valuable models that have been developed for the 100 meter sprint, and in this module, we're going to focus on one particular model, a mathematical-physical model that was developed by Mureika. And the reference is given here on this slide. So let's first present the basic equations. Well this is the first one, stating that the distance is the integral over the running time of the running speed, so with d, the distance, t, the time, capital T, the total time, and u, the running speed. And then the other equation actually expresses Newton's second law, so this is acceleration, on the left-hand side, and then on the right-hand side you see forces, actually the forces divided by mass, where we have the drive term, the maintenance term, the velocity term and the drag term. And, as mentioned before, this is actually nothing more, nothing less than Newton's second law. So, let's first look at the phases of the 100 meter dash and 100 meter sprint. Again focusing on this equation, the first phase actually is called the drive phase. So, that's about the first 25 to 30 meters, where actually the athlete starts from this crouched position and then accelerates his or her body, and that is possible due to the efficient starting position. And this phase is actually described by the drive term. Then there is a transition phase, and that's the phase where actually the body of the athlete becomes from the crouched position into a more upright position. And that's then the so-called maintenance phase that lasts about 50 meters, where the athlete actually has achieved the maximum speed and where actually also muscular fatigue is setting in. And this is described by the so-called maintenance term. So let's have a look at the two propulsive terms. So actually the terms that provide the acceleration to the athlete. First, we look at the drive term, and the drive term can be expressed by this exponential function, where you have f naught, which is the magnitude of the drive, and sigma is a constant. And typical values for top athletes, or athletes that run the 100 meter sprint at about ten seconds, or below ten seconds, are given here with the appropriate units. And this can be presented in this graph, so you can see that, of course, this drive term, which is typical for this crouched position, is large in the beginning, but then rapidly drops to a very low value. Let's have a look at the maintenance term, which is the second propulsive term, also expressed by an exponential function, as an exponential function of time, with f1 the magnitude of maintenance, c a constant. And also again typical values for sub-ten seconds athletes are given below. And here this is a graphical representation of this force divided by mass, so it's Newton per kilogram, in blue on this curve. You see that this value actually is quite high over the entire duration of the race. Let's then look at the counter-propulsive terms, so the terms that actually slow down the athlete. First the velocity term. The velocity term actually represents, what you could call the physical barrier. That limits actually the maximum speed beyond just the pure muscular strength. So it also represents actually the curtailing of the acceleration with increasing speed. However, mind you, this is not a drag term. The drag term is a different force that will be described later. And this velocity term can be expressed with this rather simple function, and the typical value for the constant alpha is also given here for a typical value for top athletes. And when we express this in a graph, it's actually this curve, the orange curve that starts from zero and then rapidly increases and then remains at a fairly high value. And then finally, there's the drag term. The drag term, which is a very important term, but it will be shown that this term, actually, compared to the others, is quite low. It's quite small. And this is the typical expression as we know it already from also the previous module, but also the previous weeks in this MOOC. The drag force, in this case, divided by the mass, M. And we took here 80 kilograms as an example for a typical sub-ten second male athlete. And if you express this in this graph you see indeed that this force, well this force divided by mass, is very low. But nevertheless, it is very important and we will show this later in this week. Actually, there is also a correction that can be applied to this drag term. And actually this is a correction taking into account the fact that the athlete indeed starts from a crouched position and then moves towards an upright position. And then actually, there's a reduction of the frontal area of this runner. But if you actually plot this correction, actually both the uncorrected and the corrected drag term, you see that actually the differences are very small. There are some differences, especially in the beginning of course, because of the start from the crouch position, but they are very limited. And then this is the summary, so we have this equation, Newton's second law, with the expression of the four forces, so the propulsive forces and the counter-propulsive forces given here. And then you can solve this. You can solve this by numerical integration. A typical time step needed here is one microsecond in order to have time-step insensitive results. Then we can use actually this numerical integration to look at some typical results. So here the typical values for a sub-ten second top male athlete are given. And then we have a curve here that represents the time as a function of distance, but also the speed as a function of the distance. And you see that indeed the speed, of course, starts from zero, then rapidly increases, reaches a maximum value and then slightly drops-off towards the end of the race. What you can also do is to determine the so-called time-velocity curve. So this is the speed, the running speed as a function of time. And also here you see that there is a maximum speed that is achieved at slighty less than six seconds and then afterwards slightly drops off. So let's turn back to the module question now, where we actually asked at which distance a top athlete that is running the 100 meter sprint will reach his or her maximum speed. And the right answer is actually at about half of the race, actually a bit more than half. And this was indeed shown by this graph that was determined based on solving these two equations by numerical integration with a small enough time step. And indeed, it's clear that there is a maximum speed here that is achieved slightly above the time of six seconds. In this module, we've learned about the various stages of the 100 meter sprint, the background of a mathematical-physical model of the 100 meter sprint, and a time-velocity curve of the 100 meter sprint. In the next module, we're going to focus on how wind effects can be implemented in this model. What the effects actually are of head winds versus tail winds, and what the effects of head and tail wind are on the resulting 100 meter sprint performance. Thank you again for watching, and we hope to see you again in the next module.