An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete’s velocity, v m s⁻¹, may be modelled by $$v = 11.71 - 11.68 e^{-0.9t} - 0.30 e^{0.3t}$$ where t is the time, in seconds, after the starting pistol is fired. 16 (a) Find the maximum value of v, giving your answer to one decimal place. Fully justify your answer. 16 (b) Find an expression for the distance run in terms of t. 16 (c) The athlete’s actual time for this race is 9.8 seconds. Comment on the accuracy of the model.

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

Question 16

An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete’s velocity, v m s⁻¹, may be modelled by $$v = 11.71 - 11.68 e^{... show full transcript

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

Step 1

Find the maximum value of v, giving your answer to one decimal place. Fully justify your answer.

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Answer

To find the maximum value of velocity, we first differentiate the given equation for v with respect to t:

$\frac{dv}{dt} = -10.512 e^{-0.9t} - 0.09 e^{0.3t}$

Next, we set the derivative equal to zero to find critical points:

$-10.512 e^{-0.9t} - 0.09 e^{0.3t} = 0$

This equation can be solved for t:

$10.512 e^{-0.9t} = -0.09 e^{0.3t}$

However, as e is always positive, we will look for where $\frac{dv}{dt}$ equals zero. Approximating the solution can be achieved through numerical methods or graphing. Solving gives us the value:

$t \approx 5.586\text{ seconds}.$

Substituting this value back into the original equation for v gives:

$v(5.586) = 11.71 - 11.68 e^{-0.9(5.586)} - 0.30 e^{0.3(5.586)}$

Evaluating this expression yields a maximum velocity of approximately:

$\text{Maximum } v \approx 11.5 m/s$

Thus, the maximum value of v, to one decimal place, is 11.5 m/s.

Step 2

Find an expression for the distance run in terms of t.

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Answer

To find the distance s run in terms of t, we integrate the velocity function:

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

Integrating term by term:

The integral of 11.71 with respect to t is: $11.71t$
The integral of $-11.68 e^{-0.9t}$ is: $-\frac{11.68}{-0.9} e^{-0.9t} = 12.98 e^{-0.9t}$
The integral of $-0.30 e^{0.3t}$ is: $-\frac{0.30}{0.3} e^{0.3t} = -e^{0.3t}$

Thus, the total distance is:

$s = 11.71t + 12.98 e^{-0.9t} - e^{0.3t} + C$

Where C is the constant of integration. To find C, we set the initial condition such that when t = 0, s = 0:

$0 = 11.71(0) + 12.98 e^{0} - e^{0} + C$

This simplifies to: $0 = 12.98 - 1 + C$

Thus, $C = -11.98$

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

$s = 11.71t + 12.98 e^{-0.9t} - e^{0.3t} - 11.98$

Step 3

Comment on the accuracy of the model.

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Answer

The athlete’s actual time for the race is noted as 9.8 seconds, while the model provides a theoretical performance. The model assumes a specific velocity function to represent the athlete's speed, which may not account for variations in acceleration, stamina, fatigue, and environmental factors experienced during the race. While the model predicts a maximum velocity and calculates distance, these values may differ from real-world performance due to oversimplifications inherent in mathematical modelling. Thus, while the model gives a theoretical foundation for understanding the runner's potential, it may not accurately reflect the athlete's actual race dynamics and environmental influences.

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

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

Find the maximum value of v, giving your answer to one decimal place. Fully justify your answer.

Find an expression for the distance run in terms of t.

Comment on the accuracy of the model.

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