Question - Estimation of Train Distance Travelled Over Time
Solution:
The image displays a graph that shows the velocity of a train over time, and the instructions are to use four strips of equal width to estimate the distance traveled in the first 20 seconds, followed by a question whether this estimation would be an underestimate or overestimate.To solve part (a):1. The first step is to divide the total time of 20 seconds into four equal time intervals, which means each interval will be 5 seconds long.2. Determine the approximate velocity of the train at each of the intervals. Since the graph is a curve rather than straight lines, we estimate the average velocity over each interval by taking a value from the graph that seems reasonable, such as the value at the midpoint of each interval.Let's say, for the sake of argument, the velocity values (in meters per second) were approximately determined from the graph as:- At 2.5 seconds: 10 m/s- At 7.5 seconds: 20 m/s- At 12.5 seconds: 30 m/s- At 17.5 seconds: 40 m/s3. For each interval, calculate the distance traveled by multiplying the average velocity by the interval time (5 seconds):- From 0 to 5 seconds: $$ 10 \text{ m/s} \times 5 \text{ s} = 50 \text{ m} $$- From 5 to 10 seconds: $$ 20 \text{ m/s} \times 5 \text{ s} = 100 \text{ m} $$- From 10 to 15 seconds: $$ 30 \text{ m/s} \times 5 \text{ s} = 150 \text{ m} $$- From 15 to 20 seconds: $$ 40 \text{ m/s} \times 5 \text{ s} = 200 \text{ m} $$4. Add up the distances from each interval to obtain the total estimated distance traveled in the first 20 seconds:- Total distance $$ = 50 \text{ m} + 100 \text{ m} + 150 \text{ m} + 200 \text{ m} = 500 \text{ m} $$For part (b):The estimation using this method is an underestimate. This is because the graph shows that the train's velocity is increasing over time, and by taking the average velocity at each midpoint, we under-represent the velocity during the second half of each interval. Therefore, the actual area under the curve (representing the true distance) is larger than the estimated area using the mid-interval velocities for these rectangles. Thus, the distance would be underestimated using this estimation method.
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