Careful measurements have been made of Olympic sprinters in the 100-metre dash. A quite realistic model is that the sprinter’s velocity is given by 𝑣𝑣𝑥𝑥 = 𝛿𝛿(1 − 𝑒𝑒−𝛽𝛽𝛽𝛽) Where t is in seconds, 𝑣𝑣𝑥𝑥 is in m s–1 and the constants 𝛿𝛿 and 𝛽𝛽 are characteristic of the sprinter. Sprinter Carl Lewis’ run at the 1987 World Championships is modelled with 𝛿𝛿 = 11.81 ms–1 and 𝛽𝛽 = 0.6887 s–1. (a) What was Lewis’ acceleration at t = 0.00 s, 2.00 s and 4.00 s? (b) Find an expression for the distance travelled at time t. (c) Your expression from part (b) is an equation that you can’t solve for t. Use ‘trial and error’ to find the time needed to travel a specific distance. To the nearest 0.01 s, find the time Lewis need 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.
Careful measurements have been made of Olympic sprinters in the 100-metre dash. A quite realistic model is that the sprinter’s velocity is given by 𝑣𝑣𝑥𝑥 = 𝛿𝛿(1 − 𝑒𝑒−𝛽𝛽𝛽𝛽) Where t is in seconds, 𝑣𝑣𝑥𝑥 is in m s–1 and the constants 𝛿𝛿 and 𝛽𝛽 are characteristic of the sprinter. Sprinter Carl Lewis’ run at the 1987 World Championships is modelled with 𝛿𝛿 = 11.81 ms–1 and 𝛽𝛽 = 0.6887 s–1. (a) What was Lewis’ acceleration at t = 0.00 s, 2.00 s and 4.00 s? (b) Find an expression for the distance travelled at time t. (c) Your expression from part (b) is an equation that you can’t solve for t. Use ‘trial and error’ to find the time needed to travel a specific distance. To the nearest 0.01 s, find the time Lewis need 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.
