Randall D. Knight — Kinematics in One Dimension: Problem Challenge_Problem_82

Careful measurements have been made of Olympic sprinters in the 100 meter dash. A quite realistic model is that the sprinter’s velocity is given by \[ v_x = a(1 - e^{-bt}) \] where \( t \) is in s, \( v_x \) 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 \( a = 11.81 \text{ m/s} \) and \( b = 0.6887 \text{ s}^{-1} \). a. What was Lewis’s acceleration at \( t = 0 \text{ s} \), \( 2.00 \text{ s} \), and \( 4.00 \text{ s} \)? b. Find an expression for the distance traveled at time \( t \). c. 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.