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The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground - Edexcel - A-Level Maths Pure - Question 4 - 2019 - Paper 2
Question 4
The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground. The results ... show full transcript
Worked Solution & Example Answer:The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground - Edexcel - A-Level Maths Pure - Question 4 - 2019 - Paper 2
Step 1
estimate the length of runway used by the jet to take off
Answer
To estimate the length of runway used, we can approximate the area under the speed-time graph using the trapezium rule. The speeds are measured at intervals of 5 seconds, so we will use the average speeds for each interval.
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Calculate the areas for each segment:
- From 0 to 5 seconds: Average speed = (2 + 5)/2 = 3.5 ms⁻¹, Area = 3.5 * 5 = 17.5 m
- From 5 to 10 seconds: Average speed = (5 + 10)/2 = 7.5 ms⁻¹, Area = 7.5 * 5 = 37.5 m
- From 10 to 15 seconds: Average speed = (10 + 18)/2 = 14 ms⁻¹, Area = 14 * 5 = 70 m
- From 15 to 20 seconds: Average speed = (18 + 20)/2 = 19 ms⁻¹, Area = 19 * 5 = 95 m
- From 20 to 25 seconds: Average speed = (20 + 42)/2 = 31 ms⁻¹, Area = 31 * 5 = 155 m
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Summing these areas gives us the total distance (length of runway):
Total Length = 17.5 + 37.5 + 70 + 95 + 155 = 375 m
Thus, the estimated length of runway used by the jet to take off is approximately 375 m.
Step 2
explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off
Answer
My answer to part (a) is an overestimate. This can be concluded because the speed of the jet was increasing throughout the 25-second interval, which means the actual speed at each interval likely exceeded the average calculated speeds. Additionally, when estimating using the trapezium rule, as all rectangles/areas representing speed are below the actual curve where the speed is increasing, it follows that the area calculated (length calculated) exceeds the actual distance traveled due to acceleration. Thus, the runway used is more accurately represented to be less than 375 m.