A velocity vs. time graph for a runner is measured at 2-second intervals. To estimate the total distance from 0 to 12 seconds, which approach should be used?

A velocity vs. time graph for a runner is measured at 2-second intervals. To estimate the total distance from 0 to 12 seconds, which approach should be used?

• A. Sum velocity values at each interval
• B. Determine the area under the velocity-time curve between 0 and 12 seconds
• C. Average velocity times 12
• D. Use the velocity at 12 seconds times 12

Answer: (B)

Distance is found by measuring how much area under the velocity–time curve accumulates over a time interval. That’s because velocity tells you how fast position is changing, and integrating velocity over time adds up all those little changes to give total distance. With measurements every 2 seconds, you estimate the total distance from 0 to 12 seconds by calculating the area under the curve across that whole span. In practice, that means summing the velocity values over each 2-second block and combining them to form the total area, which is the Riemann-sum idea: velocity times the interval width, added up across all intervals. This area-under-the-curve approach directly corresponds to distance traveled. Summing the velocity values alone misses how long each velocity lasts, so it wouldn’t give distance. Using an average velocity times the total time would only be exact if velocity were constant (or if the average properly reflects the whole stretch), which isn’t guaranteed here. Relying on the velocity at the end point or at a single moment ignores variation throughout the interval, leading to a poor estimate.

Distance is found by measuring how much area under the velocity–time curve accumulates over a time interval. That’s because velocity tells you how fast position is changing, and integrating velocity over time adds up all those little changes to give total distance.

With measurements every 2 seconds, you estimate the total distance from 0 to 12 seconds by calculating the area under the curve across that whole span. In practice, that means summing the velocity values over each 2-second block and combining them to form the total area, which is the Riemann-sum idea: velocity times the interval width, added up across all intervals. This area-under-the-curve approach directly corresponds to distance traveled.

Summing the velocity values alone misses how long each velocity lasts, so it wouldn’t give distance. Using an average velocity times the total time would only be exact if velocity were constant (or if the average properly reflects the whole stretch), which isn’t guaranteed here. Relying on the velocity at the end point or at a single moment ignores variation throughout the interval, leading to a poor estimate.