Modeling a 100m race In this project we will develop an advanced model for the motion of a sprinter during a 100m race.

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Modeling a 100m race
In this project we will develop an advanced model for the motionof a sprinter during a 100m race. We will build the modelgradually, adding complications one at a time to develop arealistic model for the race.
(a) A sprinter is accelerating along the track. Draw a free-bodydiagram of the sprinter, including only horizontal forces. Try tomake the length of the vectors correspond to the relativemagnitudes of the forces.
Let us assume that the sprinter is accelerated by a constanthorizontal driving force, F = 400N, from the ground all the wayfrom the start to the 100m line (averaged over a few steps). Themass of the sprinter is m = 80kg:
(b) Find the position, x(t), of the sprinter as a functionof time.
(c) Show that the sprinter uses t = 6.3s to reach the 100mline.
This is a bit fast compared with real races. However, realsprinters are limited by air resistance. Let us introduce a modelfor air resistance by assuming that air resistance force isdescribed by a square law: D = (1/2)ρ*C_D*A(v − w)^2 (1) where ρ isthe density of air, A is the cross-sectional area of the runner,C_D is the drag coefficient, v is the velocity of the runner, and wis the velocity of the air. At sea level ρ = 1.293kg/m3 , and forthe runner we can assume A = 0.45m^2 , and C_D = 1.2. You caninitially assume that there is no wind: w = 0m/s. Assume that therunner is only affected by the constant driving force, F, and theair resistance force, D. (d)
d) Find an expression for the acceleration of the runner.
e) Use Euler’s method to find the velocity, v(t), and position,x(t) as a function of time for the runner. The runner starts fromrest at the time t = 0s. Plot the position, velocity andacceleration of the runner as a function of time. How did youdecide on the time-step ∆t? (Your answer should include the programused to solve the problem and the resulting plots).
f) Use the results to find the race time for the 100m race.
g) Show that the (theoretical) maximum velocity of a runnerdriven by these forces is: vT = sqrt(2F/ρC_DA ). The runner mayhave to run more than 100m to reach this velocity. (We often callthis maximum velocity the terminal velocity – “terminal” becausethe velocity increases until it reaches the terminal velocity,where the acceleration becomes zero). Find the numerical value ofthe terminal velocity for the runner. Do you think this isrealistic?
So far the model only includes a constant driving force and airresistance. This is clearly a too simplified model to be realistic.Let us make the model more realistic by adding a few features.First, there is a physiological limit to how fast you can run. Thedriving force from the runner should therefore decrease withvelocity, so that there is a maximum velocity at which theacceleration is zero even without air resistance. While we do notknow the detailed physiological mechanisms for this effect, we canmake a simplified force model to implement the effect byintroducing a driving force, FD, with two terms: a constant term,F, and a term that decreases with increasing velocity, FV :
FV = −fV*v , so that the driving force is: FD = F + FV = F −fV*v . Reasonable values for the parameters are F = 400N, and fv =25.8Ns/m. (These values are chosen to make the maximum velocityreasonable – they are not based on a physiologicalconsideration).
(h) If you assume that the runner is subject only to these twodriving forces, what is his maximum velocity? (You can ignore thedrag term, D, in this calculation).