Name: estimating distance from changing velocity
Uploaded: 2026-01-12T12:24:58.000Z
Duration: 1 min 29 s
Channel: JoVE

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&#8211;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.

distance problem

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:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19974/19974_Equation_1_editor_19974.svg" alt="Limit of Riemann sum equation, formula for definite integral, mathematical concept." data-altupdatedat="1775223722000" data-alt="19974_Equation_1.svg" /></figure>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.

Distance Problem

When an object's velocity changes over time, the total distance traveled can be determined by summing small displacement intervals over short increments. ...

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JoVE Core

Calculus

Unit

Integrals

KEY TERMS AND CONCEPTS

estimating distance from changing velocity

Estimating distance from changing velocity uses small time intervals and velocity values. When an object speeds up or slows down, its total displacement can be found by adding many short pieces of motion. This method gives an approximation first, and integral calculus gives the exact result later.

A runner who accelerates during the first three seconds of a race shows how the idea works. Velocity can be measured every half-second and multiplied by the 0.5-second time step. Using left-endpoint velocities gives an estimated distance of 10.55 meters. Using right-endpoint velocities gives a higher estimate of 13.65 meters.

These two estimates show that the true distance traveled should fall between them. More frequent velocity measurements make the estimate more accurate. To find the exact displacement, the sum of velocity times small time intervals is taken in the limit as the intervals get smaller and smaller.

This limit is the Riemann sum definition of an integral. It shows that total distance traveled is the integral of velocity over time. As the time steps become infinitesimally small, the summation matches the integral and removes the error from numerical approximation.

Estimating Distance from Changing Velocity

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key terms and concepts

Learn how changing velocity can be used to estimate distance with left- and right-endpoint sums, then find the exact result using limits and integrals.

Estimating Distance from Changing Velocity | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Estimating Distance from Changing Velocity

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