Chapter 2, Problem 82EAP

III Careful measurements have been made of Olympic sprinter in the 100 meter dash. A quite realistic model is that the sprinter's velocity is given by

$v_x=a\left(1-e^{-bt}\right)$

where t is in s, v_xis 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 $b=0.6887s^{-1}$

1. What was Lewis's acceleration at t = 0 s, 2.00 s, and 4.00 s?

Find an expression for the distance traveled at time t.

Your expression from part b 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 to 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.