An elite athlete runs in a straight line to complete a 100-metre race - AQA - A-Level Maths Pure - Question 16 - 2019 - Paper 2

The distance run, s, can be found by integrating the velocity function:

$s = \int v \, dt = \int \left(11.71 - 11.68e^{-0.9t} - 0.30e^{0.3t}\right) dt$

Integrating term by term:

$s = 11.71t + 12.978e^{-0.9t} - 1.0e^{0.3t} + C$

To find the constant of integration, we use the condition that at t = 0, s = 0:

$s(0) = 11.71(0) + 12.978(1) - 1(1) + C = 0$

This gives:

$C = -12.978 + 1 = -11.978$

Thus, the expression for distance run in terms of t is:

$s = 11.71t + 12.978e^{-0.9t} - 1.0e^{0.3t} - 11.978$

The athlete's actual time for this race is 9.8 seconds, while the model's calculated maximum speed occurs at approximately 5.586 seconds, which may not match an actual race condition. While the model gives a theoretical representation of the athlete's performance, it may not account for external factors such as fatigue or acceleration phases. Therefore, while the model is useful, its accuracy in predicting the exact race outcome can vary.