When an object's velocity changes over time, the total distance traveled can be determined by summing small displacement intervals over short increments. This approach approximates the true distance through numerical summation and the use of integral calculus. An estimate of the total displacement can be obtained by measuring velocity at regular intervals and multiplying each value by the corresponding time step.
If a runner accelerates over the first three seconds of a race, speed measurements taken every half-second indicate an increase from 0 to 6.2 meters per second. One estimation method involves using left-endpoint velocities, where each velocity value is multiplied by the 0.5-second time step, producing an approximate distance of 10.55 meters.
Another method applies right-endpoint velocities, yielding a slightly higher estimate of 13.65 meters. The true distance traveled lies between these two estimates, with more frequent velocity measurements improving accuracy.
A more precise determination of distance relies on the concept of limits. The exact displacement can be found by taking the limit of the sum of velocity values multiplied by small time intervals as the number of measurements increases. Mathematically, this is expressed as:
This limit corresponds to the Riemann sum definition of the integral, demonstrating that the total distance traveled is the integral of velocity over time. As the time intervals become infinitesimally small, the summation converges precisely to the integral, eliminating the approximation error associated with numerical summation methods.
The distance problem finds how far an object has traveled using velocity measured at different points in time.
When velocity varies, the total distance can be approximated by adding small displacement intervals, each showing motion over a short time step.
For example, in a race, a runner accelerates steadily during the first three seconds. Velocity measurements taken every half-second show the speed increasing from 0 to 6.2 meters per second.
These measurements are used to estimate the total distance by dividing the velocity–time graph into half-second rectangles for the lower sum and the upper sum.
The lower estimate uses velocities at the left endpoints of each time interval. Multiplying each of these velocities by the half-second time step and summing the individual results gives 10.55 meters.
On the other hand, the upper estimate uses right endpoint velocities. Multiplying each of these by the time step and adding gives 13.65 meters. The actual distance lies between these two estimates. Increasing the number of measurements leads to a more accurate result.
With infinite measurements, the distance equals the area under the curve, shown by the integral of velocity over time.
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