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

距離問題は、異なる時間点で測定された速度を用いて物体がどれだけ遠くを移動したかを示します。速度が変化する場合、小さな変位間隔を加えることで総距離を近似でき、それぞれが短い時間ステップでの運動を示します。例えば、レースではランナーが最初の3秒間は着実に加速します。0.5秒ごとに速度測定を行うと、速度が0から6.2メートル毎秒に増加していることがわかります。これらの測定値は、速度-時間グラフを下和と上和の半秒長方形に割って総距離を推定するために用いられます。下位推定値は各時間区間の左端点の速度を用います。これらの速度にそれぞれ0.5秒の時間ステップを掛け、個々の結果を合計すると10.55メートルとなります。一方、上限推定値は右端点速度を用います。これらにそれぞれ時間ステップを掛けて足すと13.65メートルになります。実際の距離はこの2つの推定値の間にあります。測定回数を増やすことで、より正確な結果が得られます。無限大の測定では、距離は曲線下面積に等しく、時間に従った速度の積分で示されます。

距離の問題

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Education

JoVE Core

Calculus

微積分学

Unit

単位

Integrals

積分

KEY TERMS AND CONCEPTS

主要用語と概念

Distance Problem

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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