An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete's velocity, v ms⁻¹, may be modelled by $$v = 11.71 - 11.68e^{-0.9t} - 0.30e^{0.3t}$$ 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.

Photo AI

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 ms⁻¹, may be modelled by $$v = 11.71 - 11.68e^{-0... 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.

96%

114 rated

Answer

To find the maximum value of the velocity function, we differentiate the velocity equation with respect to time t:

$\frac{dv}{dt} = -11.68 \cdot (-0.9)e^{-0.9t} + 0.30 \cdot (0.3)e^{0.3t}$

Setting the derivative equal to zero for maximization:

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

Rearranging gives:

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

This equation needs to be solved for t. After substituting values and trial and error, we find:

$t \approx 5.586$ seconds.

Next, we evaluate the original velocity function at this value:

$v = 11.71 - 11.68e^{-0.9 \cdot 5.586} - 0.30e^{0.3 \cdot 5.586}$

Calculating gives:

$v \approx 11.5 \, \text{ms}^{-1}$

Thus, the maximum value of velocity v is approximately 11.5 ms⁻¹.

Step 2

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

99%

104 rated

Answer

To find the distance run, we integrate the velocity function with respect to time t:

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

Integrating gives:

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

where C is the constant of integration. Applying the initial condition, we set $s = 0$ when $t = 0$ , which leads to:

$C = -12.878$

Combining this, we arrive at the expression for the distance run:

$s = 11.71t + 12.978e^{-0.9t} - 1.0e^{0.3t} - 12.878$

Step 3

Comment on the accuracy of the model.

96%

101 rated

Answer

Given that the athlete's actual time for the race is 9.8 seconds, we can assess the accuracy of the velocity model. The model predicts a maximum velocity of approximately 11.5 ms⁻¹, but does not account for factors such as fatigue, acceleration and deceleration phases, and variations in speed during the race. Therefore, while the model provides a theoretical maximum velocity, it may not accurately represent the athlete's performance over the entire race distance of 100 meters.

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.

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

Comment on the accuracy of the model.

Join the A-Level students using SimpleStudy...