Name: estimating distance from velocity data
Uploaded: 2026-01-12T12:24:45.000Z
Duration: 1 min 26 s
Channel: JoVE

Imaginez un camion circulant sur une autoroute. Pour déterminer la distance parcourue durant un intervalle de temps donné t, on trouve l’aire sous son graphique vitesse–temps.Comme la vitesse du camion est connue à des points temporels discrets, cette zone ne peut pas être calculée précisément. Au lieu de cela, la surface est approximée à l’aide de la règle du trapèze. Pour ce faire, divisons l’intervalle total de temps en n sous-intervalles égaux, chacun ayant une largeur de Δt.Dans cette méthode, les deux extrémités de chaque sous-intervalle sont reliées par une ligne droite, formant des trapèzes.Pour un sous-intervalle donné, qui possède deux faces verticales représentant les vitesses et une base horizontale représentant la largeur du temps, la surface de ce trapèze est la moyenne des longueurs de ses deux côtés verticaux multipliée par la distance horizontale.Ajouter les aires de tous les n trapèzes donne une approximation du déplacement.Après manipulations algébriques et résolution de cette somme, la première et la dernière vitesse apparaissent une fois, tandis que les vitesses intérieures apparaissent deux fois car chacune appartient à deux trapèzes.Après avoir combiné des termes similaires, l’expression résultante approxime le déplacement total sur l’intervalle de temps spécifié.

méthodes du point milieu et des trapèzes

Estimer la distance parcourue par un v&#xE9;hicule &#xE0; partir de sa vitesse enregistr&#xE9;e en fonction du temps est un probl&#xE8;me courant en physique et en ing&#xE9;nierie. Lorsque les donn&#xE9;es de vitesse sont disponibles &#xE0; intervalles de temps discrets, plut&#xF4;t que sous forme de fonction continue, des m&#xE9;thodes d&#x2019;int&#xE9;gration num&#xE9;rique telles que la r&#xE8;gle des trap&#xE8;zes sont souvent utilis&#xE9;es pour estimer le d&#xE9;placement total.
La r&#xE8;gle des trap&#xE8;zes consiste &#xE0; diviser l&#x2019;intervalle de temps total en plusieurs segments &#xE9;gaux. Dans chaque segment, les vitesses enregistr&#xE9;es aux extr&#xE9;mit&#xE9;s sont reli&#xE9;es par une droite, formant un trap&#xE8;ze sur un graphique vitesse&#x2013;temps. Chaque trap&#xE8;ze poss&#xE8;de un c&#xF4;t&#xE9; vertical repr&#xE9;sentant la vitesse au d&#xE9;but et &#xE0; la fin du segment, ainsi qu&#x2019;une base horizontale correspondant &#xE0; l&#x2019;intervalle de temps.
Pour estimer le d&#xE9;placement, on calcule l&#x2019;aire de chaque trap&#xE8;ze. Cette aire correspond &#xE0; la moyenne des vitesses aux extr&#xE9;mit&#xE9;s du segment, multipli&#xE9;e par la dur&#xE9;e du segment. La somme des aires de tous les trap&#xE8;zes fournit une estimation de la distance totale parcourue sur toute la p&#xE9;riode.
Dans ce processus, les vitesses enregistr&#xE9;es au d&#xE9;but et &#xE0; la fin de l&#x2019;intervalle de temps total ne sont utilis&#xE9;es qu&#x2019;une seule fois, tandis que les vitesses aux points interm&#xE9;diaires interviennent dans deux trap&#xE8;zes adjacents. Cette approche syst&#xE9;matique permet une approximation fiable du d&#xE9;placement, notamment lorsque le nombre d&#x2019;intervalles est suffisamment grand. La r&#xE8;gle des trap&#xE8;zes est donc une technique fondamentale en analyse num&#xE9;rique, offrant un moyen pratique d&#x2019;int&#xE9;grer des donn&#xE9;es empiriques issues du mouvement ou d&#x2019;autres processus d&#xE9;pendant du temps.

Méthodes du point milieu et des trapèzes

Estimer la distance parcourue par un v&#xE9;hicule &#xE0; partir de sa vitesse enregistr&#xE9;e en fonction du temps est un probl&#xE8;me courant en ...

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Techniques of Integration

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Trapezoidal Rule

estimating distance from velocity data

Estimating distance from velocity data is a common use of the trapezoidal rule. In physics and engineering, velocity is often recorded at separate time points instead of as one continuous equation. Numerical integration gives a way to approximate the total displacement from those measurements.

The method divides the full time interval into equal parts. For each part, the velocities at the two endpoints are joined with a straight line on a velocity-time graph. That line creates a trapezoid. Each trapezoid has a vertical side for velocity and a horizontal base for the time interval.

To estimate displacement, the area of each trapezoid is found. This area depends on the average of the endpoint velocities and the length of the time segment. Adding the areas of all the trapezoids gives an estimate of the total distance traveled over the entire period.

The first and last velocity values are used only once. The values in the middle are shared by two neighboring trapezoids. This makes the method efficient for discrete data. It is especially useful when the number of intervals is large, and it provides a practical way to integrate measured motion data and other time-dependent processes.

Estimating Distance from Velocity Data

integration by parts: indefinite integrals

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Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula. The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle...

trapezoidal rule

Estimating distance from velocity data is a common use of the trapezoidal rule. In physics and engineering, velocity is often recorded at separate time points instead of as one continuous equation. Numerical integration gives a way to approximate the total displacement from those measurements. The method divides the full time interval into equal parts. For each part, the velocities at the two endpoints are joined with a straight line on a velocity-time graph. That line creates a trapezoid.

simpson's rule i

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key terms and concepts

Learn how the trapezoidal rule uses velocity-time data to estimate displacement by splitting time into intervals and adding trapezoid areas.

Estimating Distance from Velocity Data | 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 Velocity Data

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