Careful measurements have been made of Olympic sprinters in the 100-meter dash.
ID: 1686170 • 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 byv_x = a{1 - e^{ - bt} })
Sprinter Carl Lewis's run at the 1987 World Championships is modeled with a = 11.81 m/s and b = 0.6887{ s}^{ - 1}.
a)What was his time at t=0s.
b)What was his time at t=2.35s
c)What was his time at t=4.35s
d)Find an expression for the distance traveled at time t.
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 { m{ s}}, find the time Lewis needed to sprint 100.0 { m m}. His official time was 0.01 { m s} more than your answer, showing that this model is very good, but not perfect
Explanation / Answer
a)The sprinter's velocity is given by v_x = a{1 - e^(-bt)} where a = 11.81 m/s and b = 0.6887 s^-1 The sprinter's velocity at t = 0 s is v(0) = a{1 - e^(0)} = a{1 - 1} = 0 b)The sprinter's velocity at t = 2.35 s is v(2.35) = 11.81 * {1 - e^(-0.6887 * 2.35)} c)The sprinter's velocity at t = 4.35 s is v(4.35) = 11.81 * {1 - e^(-0.6887 * 4.35)} d)The expression for the distance travelled at time t is v_x = a{1 - e^(-bt)} or (dx/dt) = a{1 - e^(-bt)} or dx = a{1 - e^(-bt)} * dt Integrating the above equation,we get x = (a/b) * e^(-bt) + at