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A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions - Edexcel - A-Level Maths Pure - Question 11 - 2019 - Paper 2
Question 11
A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions. The engineer measures and records the braking d... show full transcript
Worked Solution & Example Answer:A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions - Edexcel - A-Level Maths Pure - Question 11 - 2019 - Paper 2
Step 1
Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula
Answer
The linear relationship observed in Figure 6 indicates that when the logarithm of the braking distance, log<sub>10</sub>d, is plotted against the logarithm of the speed, log<sub>10</sub>V, the data aligns closely with a straight line. This suggests that the braking distance can be expressed in a power-law form, which is mathematically represented as d = kV<sup>n</sup> for constants k and n. Specifically, the slope of the line gives a value of n that represents the relationship's dependency on speed. In the context of the data point (0, -1.77), where d = 20 and V = 30, we can confirm the model structure aligns with the collected evidence.
Step 2
Using the information given in Figure 5, with k = 0.017
Answer
Given d = 20 when V = 30, we substitute these values into the model:
20 = 0.017 * (30)<sup>n</sup>.
We then solve for n:
First, rearrange:
n = log<sub>10</sub>(20 / 0.017) / log<sub>10</sub>(30).
Calculating logarithms:
20 / 0.017 ≈ 1176.47, log<sub>10</sub>(1176.47) ≈ 3.071,
and log<sub>10</sub>(30) ≈ 1.477.
Thus, we have:
n ≈ 3.071 / 1.477 ≈ 2.08 (rounded to 3 significant figures gives 2.08).
Step 3
Use your formula to find out if Sean will be able to stop before reaching the puddle.
Answer
First, calculate the total distance traveled by Sean during the reaction time:
Distance = Speed × Time = 60 kmh<sup>-1</sup> × (0.8 s × (1/3600) h/s) ≈ 0.01333 km or 13.33 m.
Thus, the car would cover 100 m (distance to puddle) - 13.33 m (reaction distance) = 86.67 m left to stop.
Now applying the formula for stopping distance:
d = kV<sup>n</sup>, with k = 0.017 and V = 60.
d = 0.017 * (60)<sup>2.08</sup>.
Calculating:
60<sup>2.08</sup> ≈ 65.08, d = 0.017 * 65.08 ≈ 1.10736 m.
Since 1.10736 m < 86.67 m, Sean will stop before reaching the puddle.