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Here is a speed-time graph for a train - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 3
Question 18
Here is a speed-time graph for a train. (a) Work out an estimate for the distance the train travelled in the first 20 seconds. Use 4 strips of equal width. (b) Is ... show full transcript
Worked Solution & Example Answer:Here is a speed-time graph for a train - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 3
Step 1
Work out an estimate for the distance the train travelled in the first 20 seconds. Use 4 strips of equal width.
Answer
To estimate the distance travelled by the train from the speed-time graph, we can break the first 20 seconds into 4 equal intervals of 5 seconds each.
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Identify the Speed for Each Interval: We will calculate the speed at the following times:
- At 0 seconds, the speed is approximately 0 m/s.
- At 5 seconds, the speed is approximately 12 m/s.
- At 10 seconds, the speed is approximately 18 m/s.
- At 15 seconds, the speed is approximately 22 m/s.
- At 20 seconds, the speed reaches approximately 25 m/s.
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Calculate the Areas under the Curve: The area under the speed-time graph represents the distance travelled. For each strip, we will find the average speed over the interval and multiply it by the interval width (5 seconds):
- For the first strip (0 to 5 seconds): Average speed = (0 + 12)/2 = 6 m/s, Area = 6 m/s * 5 s = 30 m.
- For the second strip (5 to 10 seconds): Average speed = (12 + 18)/2 = 15 m/s, Area = 15 m/s * 5 s = 75 m.
- For the third strip (10 to 15 seconds): Average speed = (18 + 22)/2 = 20 m/s, Area = 20 m/s * 5 s = 100 m.
- For the fourth strip (15 to 20 seconds): Average speed = (22 + 25)/2 = 23.5 m/s, Area = 23.5 m/s * 5 s = 117.5 m.
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Find the Total Estimated Distance: Total distance = 30 m + 75 m + 100 m + 117.5 m = 322.5 m. Thus, the estimated distance the train travelled in the first 20 seconds is approximately 322.5 meters.
Step 2
Is your answer to (a) an underestimate or an overestimate of the actual distance the train travelled? Give a reason for your answer.
Answer
The answer to part (a) is an overestimate of the actual distance the train travelled.
Reason: The method used to estimate the distance involved using trapezoidal areas to approximate the area under the curve. Because the speed of the train is constantly increasing (the curve is concave upwards), the actual distance travelled will be less than the area calculated using the trapezoids. This means our approximation overestimates the actual distance, as it assumes the speed is constant across each interval rather than accounting for the rise in speed during the intervals.