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

Le problème de la distance détermine la distance parcourue par un objet en utilisant la vitesse mesurée à différents moments du temps.Lorsque la vitesse varie, la distance totale peut être approximée en ajoutant de petits intervalles de déplacement, chacun montrant le mouvement sur un court pas de temps.Par exemple, lors d’une course, un coureur accélère régulièrement pendant les trois premières secondes. Les mesures de vitesse prises toutes les demi-secondes montrent que la vitesse augmente de 0 à 6,2 mètres par seconde.Ces mesures servent à estimer la distance totale en divisant le graphique vitesse–temps en rectangles d’une demi-seconde pour la somme inférieure et la somme supérieure.L’estimation inférieure utilise les vitesses aux extrémités gauches de chaque intervalle de temps. En multipliant chacune de ces vitesses par le pas de temps d’une demi-seconde et en additionnant les résultats individuels, on obtient 10,55 mètres.En revanche, l’estimation supérieure utilise les vitesses d’extrémité droite. En multipliant chacun de ces éléments par l’étape de temps et en additionnant, on obtient 13,65 mètres. La distance réelle se situe entre ces deux estimations. Augmenter le nombre de mesures conduit à un résultat plus précis.Avec des mesures infinies, la distance est égale à l’aire sous la courbe, indiquée par l’intégrale de la vitesse dans le temps.

problème de distance

Lorsqu&#x2019;un objet voit sa vitesse varier au cours du temps, la distance totale parcourue peut &#xEA;tre d&#xE9;termin&#xE9;e en additionnant de petits d&#xE9;placements sur de courts intervalles de temps. Cette approche permet d&#x2019;approximer la distance r&#xE9;elle par sommation num&#xE9;rique et par le recours au calcul int&#xE9;gral. Une estimation du d&#xE9;placement total peut &#xEA;tre obtenue en mesurant la vitesse &#xE0; intervalles r&#xE9;guliers et en multipliant chaque valeur par le pas de temps correspondant.Si un coureur acc&#xE9;l&#xE8;re au cours des trois premi&#xE8;res secondes d&#x2019;une course, des mesures de vitesse effectu&#xE9;es toutes les 0,5 s indiquent une augmentation de 0 &#xE0; 6,2 m/s. Une premi&#xE8;re m&#xE9;thode d&#x2019;estimation consiste &#xE0; utiliser les vitesses &#xE0; l&#x2019;extr&#xE9;mit&#xE9; gauche de chaque intervalle&#x2009;; chaque valeur de vitesse est alors multipli&#xE9;e par le pas de temps de 0,5 s, ce qui conduit &#xE0; une distance approximative de 10,55 m.Une autre m&#xE9;thode repose sur l&#x2019;utilisation des vitesses &#xE0; l&#x2019;extr&#xE9;mit&#xE9; droite de chaque intervalle, ce qui fournit une estimation l&#xE9;g&#xE8;rement plus &#xE9;lev&#xE9;e de 13,65 m. La distance r&#xE9;elle parcourue se situe entre ces deux estimations, et une fr&#xE9;quence de mesure plus &#xE9;lev&#xE9;e permet d&#x2019;am&#xE9;liorer la pr&#xE9;cision.Une d&#xE9;termination plus pr&#xE9;cise de la distance repose sur la notion de limite. Le d&#xE9;placement exact peut &#xEA;tre obtenu en prenant la limite de la somme des valeurs de vitesse multipli&#xE9;es par des intervalles de temps infinit&#xE9;simaux, lorsque le nombre de mesures tend vers l&#x2019;infini. Math&#xE9;matiquement, cela s&#x2019;exprime comme suit :\begin{equation*}d = \lim_{n \to \infty} \sum_{\textit{i}=1}^{\textit{n}} f(t_{i-1})\,\Delta t = \lim_{n \to \infty}\sum_{\textit{i}=1}^{\textit{n}} f(t_i)\,\Delta t\end{equation*}\end{document}&#xA0;Cette limite correspond &#xE0; la d&#xE9;finition de l&#x2019;int&#xE9;grale comme somme de Riemann, ce qui montre que la distance totale parcourue est l&#x2019;int&#xE9;grale de la vitesse par rapport au temps. Lorsque les intervalles de temps deviennent infinit&#xE9;simaux, la somme converge exactement vers l&#x2019;int&#xE9;grale, &#xE9;liminant ainsi l&#x2019;erreur d&#x2019;approximation associ&#xE9;e aux m&#xE9;thodes de sommation num&#xE9;rique.

Problème de distance

Lorsqu&#x2019;un objet voit sa vitesse varier au cours du temps, la distance totale parcourue peut &#xEA;tre d&#xE9;termin&#xE9;e en additionnant de ...

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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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