Calculating Usain Bolt's Power

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Information about Calculating Usain Bolt's Power

Published on January 11, 2010

Author: harband


Calculating Usain Bolt’s Power : Calculating Usain Bolt’s Power Applying calculus to a real-world example Calculus Made Easy Introduction The Fastest Man on Earth : info The Fastest Man on Earth How Powerful is Usain? : You are about to find out! How Powerful is Usain? The Steps : The Steps Obtain actual measurements of Usain’s speed Get an empirical formula for speed that fits the measurements Identify the relevant laws of physics Derive the formula for speed from the physical laws Get the value of the power by comparing the empirical formula with the derived one. Calculate the air resistance Here are the steps we'll be using to calculate the power. First, we obtain actual data of Usain’s speed from the International Association of Athletics Federations, the IAAF. We'll construct an empirical formula for Usain's speed by fitting a curve to the actual data. We identify the laws of physics relevant to a person running. Then, we'll derive the formula for speed from the physical laws. Finally, we'll get the value of Usain's power by comparing the empirical formula for speed with the derived formula. In addition, we'll calculate the force on Usain due to the air resistance. Actual Data : Actual Data Here are the results of the speed measurements made by a LIDAR device, a very precise laser range-finder. The red line shows the average speed vs the distance. The graph can be divided into two sections: First, an acceleration section in which the speed increases rapidly for the first 20 meters and then levels off as air resistance takes effect, and second, a constant speed section, in which Usain's propulsive force is just balanced by the air resistance. Fitting a Curve : curve fitting V = 4.163 * x 1/3 Fitting a Curve We get an empirical formula for the speed in the acceleration phase by fitting a curve to the points on the graph. The resulting formula shows that the speed depends on the cube root of the distance. Verifying the Fit : Closest Fitting Curve: V = 4.163 * x 1/3 curve fitting Verifying the Fit The green line on this graph shows the empirical curve, superimposed on the original curve. It is a good fit for lower speeds, but starts to deviate as the speed approaches its peak value. Physical Laws of Running - I : Physical Laws of Running - I Newton’s 2nd Law of Motion F = m × a m Now we'll derive the theoretical formula for the speed. First, let's present the physical laws of running that we'll use. The first physical law is Newton's second law of motion. When a force F, is applied to a object with mass m, the object accelerates with an acceleration A, which is proportional to the force. Applying this equation to Usain's movement: the force exerted by the ground on Usain, (the reaction of his pushing on the ground, from Newton's third law) causes him to accelerate. according to his mass. Physical Laws of Running - II : Physical Laws of Running - II The Power Equation P = F × V The second physical law we need is the equation for the power. When a force F, causes an object to move at a certain speed V, the force is said to generate power, P. The power is calculated according to the equation P = F times V. Applying this formula to Usain: the power he generates is equal to the force the ground exerts on him, times his speed. Basic Assumption – Constant Power : P = F x V = Constant constant p Basic Assumption – Constant Power We assume that Usain generates constant power over the entire length of his run, an assumption frequently made in the physics of running. Constant power means that as his speed increases, the force he exerts decreases proportionally. Deriving the Speed Formula : F = m a F V = m a V P = m a V V dV/dt = P / m P/m = z (zip) V = [2 z]1/2 t1/2 X =(8 z / 9)1/2 t 3/2 V = [3 z ] 1/3 x 1/3 Deriving the Speed Formula Now we'll derive the theoretical formula for speed as a function of x, using differential and integral calculus. Start with the 2nd law of motion: F = m A Multiply both sides of the equation by V, getting: F V = m A V The left side of the equation FV is just the power, P, which is assumed constant. So we get: P = m A V Since A is the derivative of V with respect to time, we get a differential equation for V : V dV/dt = P over m The ratio of power to mass on the right side is the constant z , the zip value. Integrating both sides of the differential equation from 0 to t, and using the initial condition v = 0 when t = 0, we get: V = the square root of 2 z times the square root of t. This is the formula for the speed as a function of time. Integrating once again from 0 to t, and using the initial condition x = 0 when t = 0, we get the formula for the distance as a function of time: X = the square root of 8 z over 9 times t to the three halves power. Solving for t as a function of x, and substituting it in the formula for V, we get V as a function of x to the one third power. This is the formula that we will compare with the empirical one. Comparing the Formulas : Comparing the Formulas From the empirical data we had arrived at this formula: and, from the laws of motion we derived this formula. Comparing them, we see that both formulas are are functions of x to the third with constant coefficients, so the coefficients must be equal to one another. Equating the coefficients of x to the third, we get an equation for z. Solving for the Power : Z = 24.058 P = m z = 90 * 24.058 P = 2165 w P = 2.9 hp Solving for the Power Now we are in a position to solve for Usain's power. Solving the previous equation for z, we get: Z = 24.058 Recalling the definition of the zip, z = P over m, we finally get the equation for the power: P = m z Substituting Usain's mass, 90 kilograms, for m: We get Usain's power to be 2,165 watts which, after dividing by 746, is 2.9 horsepower. Equivalent Horsepower : = Equivalent Horsepower Usain's power is equal to one two three, horses. That's power Calculating the Air Resistance : FU = FA FU = P / V = 2165 / 12.3 FU = 180 n FA = 180 n FA = .5 ρ V2 C A ρ = 1.2 kg/m3 V = 12.3 m/s A = 0.5 m2 C = 3.97 Calculating the Air Resistance We now have enough information to calculate the air resistance and the drag coefficient. In the constant speed section of the run, Usain's force is constant and is counterbalanced by the force due to air resistance: FU = FA. We'll obtain FA by calculating FU. Recalling that the power P is constant and is equal to force F times speed V, then the force F is the power P over the speed V. Substituting the known constant values, we get 2165 over 12.3. or a force of 180 newtons And, from the first equation above, this is equal to the value of the air resistance force. To get the drag coefficient C, we use the general formula for the force on an object due to air resistance: FA = .5 rho V squared Cee A where rho is the atmospheric density: 1.2 at 20 degrees celsius V is the speed, which = 12.3 meter per second A is Usain's cross sectional area = 0.5 meter squared Substituting in the first equation above, the drag coefficient C = 3.97. This is much higher than the drag coefficient of a body at rest with a fluid flowing around it, which is approximately equal to one. The explanation for the higher drag force seems to be the turbulence created by the motion of Usain's arms and legs. In addition, Usain has pulled ahead and so is working against the air by himself with no help from the others. Tuval Software : Tuval Software

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