Careful measurements have been made of Olympic sprinters in the 100-meter dash.
ID: 1493969 • Letter: C
Question
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 m/s and b = 0.6887s^-1. Find an expression for the distance traveled at time t. Express your answer in terms of the variables a, b, and t. x(t) = at- a/b(1-e^-bt) Your expression from part D 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.Explanation / Answer
v = 11.81(1 - e^(-0.6887t))
v = ds/dt
ds = 11.81(1-e^(-0.6887t))dt
INtegrating
s = 11.81[t + e^(-0.688t)/0.688] - 11.81[0 + 1/0.688] = 11.81[t + [e^(-0.688t) - 1]/0.688 ]
for (t = 1), s = 3.271 m
for (t = 10), s = 100.951m
for (t = 9.9), s = 99.7722
for (t = 9.95), s = 100.362
for (t = 9.93), s = 100.126