Careful measurements have been made of Olympic sprinters. in the 100-meter dash.
ID: 2054622 • Letter: C
Question
Careful measurements have been made of Olympic sprinters. in the 100-meter dash. A quite realistic model is that. it the sprinter's velocity is given by vx = a(1 - e-M) where t is in s. vx is in m/s. and the constants a and b are characteristic of the sprinter. Sprinter Carl Lewis's run at the 1987 World Championships is modeled with a = 11 81 m/s and b = 0.6887 s-1. What was Lewis's acceleration at t = 0 s, 2.004 and 4.00 s Find an expression for the distance traveled at time t. Your expression from part h is a transcendental equation, meaning that you can't solve it for t. However it's not hard to use trial and error to find the time needed to travel a specific distance. To the nearest 0.01 s. find the time Lewis needed. It's sprint 100.0 m. His official time was 0.01 s more than your answer, showing that this model is very good but not perfect. A sprinter can accelerate with constant acceleration for 4.0 s before reaching top speed. He can run the 100 meter dash in 10s.Explanation / Answer
a) acceleration = dv/dt = a(be^(-bt)) acceleration at t = 2 will be 11.81 * .6887 e^(-.6887*2) = 2.051 m/s^2 acceleration at t = 4 will be 11.81 * .6887 e^(-.6887*4) = .517 m/s^2 b)velocity = dx/dt so dx/dt = a(1-e^(-bt)) integrating from t=0 to t = t we get x = a(t - (e^(-bt)/-b)) from 0 to t x = a(t + (e^(-bt)/b)) from 0 to t we get x = a(t + (e^(-bt)/b)) - (a/b) c)Now given x = 100 m last part we have to use higher algebra. I will give you the hint how to do that. expand exponential series. Higher order terms will be neglected as it is given that time is very less just greater that .01 so the higher order terms in the expansion can be neglected as compared to other. Solve from there to get the answer. I am not calculating, hope you can solve yourself if it is difficult feel free to contact me