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Mathematical Modeling Of Energy-Saving Mechanism For Running

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Energy is required for every mechanical and non-mechanical process in this entire universe. When a terrestrial mobilization is involved it requires optimum energy levels to ensure that the mobilization should be completed within a given time-frame. The topic of my Investigation involves mathematical Modeling of optimum energy levels for a successful completion of a given marathon race that requires endurance and stamina. This mathematical model involves limited knowledge of relationships between the mechanical performances of human muscles and its process of metabolic energy consumption.

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My model involves a lot of human postures that are involved during the marathon race and my investigation is based on the assumption that human muscles that are optimally adapted for task of efficient running should do positive work with constant efficiency. My investigation involves a lot of human biological factors and I didn’t had any prior knowledge of these factors and I didn’t have Biology as a subject for my IB Diploma hence to serve my purpose I have taken care of my cousin sister who is working as Physiotherapist who is specialized in taking care of sport injuries in India. Her valuable guidance has helped me to successfully model the optimum Energy Efficiency model to finish a marathon efficiently.

Firstly I have mentioned the essential Background Knowledge which is indeed required for better understanding of my mathematical model as it involves a lot of human biology factors which even I wasn’t aware of when I had this idea in my Mind. I have then introduced kinematics of Variable motion fused with human biological factors. My mathematical model involves a few varying and constant forces acting on the runner while running and those between the runner and the track. I have applied work-energy theorem to mathematically model the work done and energy consumed by the runner while running. The major conclusion that I have derived from my mathematical model is that when human muscles are optimally adapted to the task of marathon running the metabolic energy consumed would be a constant multiple of the positive work performed by the muscles while running. Lastly I have mentioned a list of sources of the errors that might affect the efficiency of my mathematical model and I have mentioned a list of various resources that I have used for the research purpose for my investigation.

Ever since I lost my winning position in a marathon of 5 kilometers my mind gets involved in a deep thought process to evaluate my entire running mechanism …upon deep evaluation I realized that despite of the fact that I prepared very hard for my marathon race and I put my heart in order to win the marathon and then when the D-day came everything was going as per my plans the first three laps were in time and I knew that I will be the winner but something seriously went wrong during the last 400-500 meters of my race that I lost my aim and stood 7th in the marathon out of total of 20 participants. I was way too exhausted at the end of the race and had major dehydration issues. Few days later I realized that it was the energy-efficiency that I didn’t take care of and lost the marathon race. Even that defeat inspired me or rather challenged me if I can model optimum energy levels mathematically for a given length of racing track and human biological factors and environmental factors. I then decided to choose Mathematical Modeling for Optimum Energy process for a given marathon as topic of my Investigation. Muscular activities involve consumption of metabolic energy. Muscles can still be active without performing any work; they do if they are shorten while tension is applied to muscles. When muscles are contracted isometrically (applying tension on muscles without any change in length of muscles) no work is said to be done by muscles but when muscles expands when a tension is applied to the muscles a negative work is said to be done converting mechanical energy to heat and a lot of metabolic energy is consumed in this case as compared to the case when no work is done by muscles.

Defined as ratio of the force exerted on cross-sectional area of active muscles and amount of Energy consumed in isometric contractions Value of is found to be larger for active muscles involved in faster activities as compared to inactive muscles[footnoteRef: 2] and inversely proportional to maximum shortening speed.


While applying economy of muscles ( to muscular contractions when length of muscles changes as well as isometric activities; when muscles are shortened they perform some work and they exert less forced when they are contracted isometric-ally but they have high metabolic rate and thus from the above equation their economy is lowered and when muscles are stretched they perform negative work and the force exerted on muscles rises and metabolic rate falls and economy increases.

I have researched a lot on a concrete mathematical relationship between metabolic rate and shortening speed but unfortunately there isn’t any concrete mathematical relationship between the metabolic rate and shortening speed and hence for the sake of simplicity of my mathematical model I have assumed that Economy of muscles falls linearly with increasing shortening speed and when shortening speed acquires a peak value the Economy of muscles approaches to zero.

Mathematically: Substituting Equation (1) in Equation (2): The rate of shortening of muscles ( is product of length time’s shortening speed of musclesThe efficiency of muscles is simply the ration of output power) and input power Just like any other machine I am assuming that efficiency of the muscles is also less than 100% and hence When the efficiency of muscles will be maximum this is Optimum Muscles Properties: In case of terrestrial locomotion a little amount if work is done by muscles. Time course of rate of shortening speed of the muscles Let the muscles of the foot remain in contact with ground for a time interval from to as seen in the above figure and during this time interval lets say that the varying force will rise and fall in a cosine manner and can be best expressed using a cos behavior. The shortening speed can be thus expressed in a sinusoidal manner as: From the above equations it’s pretty obvious that muscles starts expanding at the rate, when the foot hits the ground the muscles reaches a maximum length at the mid-point of the time-interval of contact of foot with ground and when the foot is lifted from the ground and the muscles starts shortening at and a force is acted upon the muscles of the foot.

The metabolic rate of this muscle at the time ‘t’ during the period of contact of the foot with the ground is given by: However from the Equation Number (3), (5) and (6) the metabolic Energy consumed consumed during the period of the contact is given as: Applying appropriate differential calculus and differentiating the above expression and applying differentiation rules for maxima it can be further found out that metabolic energy consumed will be minimum when and substituting when in the metabolic energy consumed can be mathematically expressed as: Integrating Equation (5) with respect to T the average Force during period of contact is: Assuming that muscle exerts same pattern of force while maintaining constant length the metabolic energy consumed during the period of the time will be: Hence from the above equation it’s become clear that the predicted energy consumption for an optimally adapted muscle is 1. 4 times the metabolic energy consumed in an isometric contraction assuming same pattern of force exerted and multiplying by some constant factor.

Results and Conclusions:

My Investigation was aimed at mathematical modeling of optimum energy levels while walking and running of human beings that allows human to consume lesser metabolic energy so that they can maintain their energy levels in races particularly like marathons which requires endurance stamina which can only come by pursuing an optimum energy level while running. I was successfully able to establish a mathematical model which is based on set of certain logical assumptions. The major conclusion that I have derived from my mathematical model is that when human muscles are optimally adapted to the task of marathon running the metabolic energy consumed would be a constant multiple of the positive work performed by the muscles while running. Work done by the hip and the knee and the total of moments of forces acting on hip and knee can be minimized by keeping the ground force in line with the hip. This will further lead to lower consumption of metabolic energy for efficient running. I have mathematically proven that no work is said to be done by the hip joint during strait. I have mathematically concluded that the rimless wheel gait model will consume less Metabolic energy at lower speed at it wouldn’t be energy efficient at higher speeds.


One of the major limitation of my mathematical model is that it has been modeled theoretically and I haven’t tested my model on real scenarios, although logically model seems to be correct but I have modeled a real life scenario that involves developing optimum-energy levels during a marathon or running but the validation of the mathematical model can only be verified once its applied to the real scenarios. My mathematical model involves a lot many assumptions which I am assuming them to be true but it’s not always the case that each and every assumption will hold true irrespective of the given conditions in such case my mathematical model might be questionable.

My mathematical model involves a lots of human body dynamics and since I don’t have Biology or any relevant subject as core subject of my International Baccalaureate Diploma Program hence I might have not taken all the necessary bio-mechanics factors into consideration although my cousin sister who is specialized in this domain of sports mechanics particularly sports mechanics had helped me out of her limits but I still personally feels that having prior knowledge of human body dynamics would have been an additive advantage. Before the rimless wheel gait Model I have assumed that the hip height above the ground remains constant for the sake of simplicity of mathematical model but in real world this might not always be true. I have also not taken terrain factors and environmental factors into account such as air-resistance into my mathematical model and nowadays professional athletes do take these factors with serious consideration into their professional trainings.

18 May 2020

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