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

Immagina un camion che si muove lungo un'autostrada. Per trovare la distanza che percorre durante un dato intervallo di tempo t, si trova l'area sotto il suo grafico velocità–tempo.Poiché la velocità del camion è nota in punti temporali distinti, quest'area non può essere calcolata con precisione. Invece, l'area viene approssimata usando la regola trapezoidale. Per farlo, si divide l'intervallo totale di tempo in n sottointervalli uguali, ciascuno con una larghezza di Δt.In questo metodo, i due estremi di ciascun sottointervallo sono collegati da una linea retta, formando trapezi.Per un dato sottointervallo, che ha due lati verticali che rappresentano le velocità e una base orizzontale che rappresenta la larghezza temporale, l'area di questo trapezio è la media delle lunghezze dei suoi due lati verticali moltiplicate per la distanza orizzontale.Sommando le aree di tutti i n trapezi si ottiene un'approssimazione dello spostamento.Dopo manipolazioni algebriche e la risoluzione di questa somma, la prima e l'ultima velocità appaiono una volta, mentre le velocità interne appaiono due volte perché ciascuna appartiene a due trapezi.Dopo aver combinato termini simili, l'espressione risultante approssima lo spostamento totale nell'intervallo di tempo specificato.

regola del punto medio e regola dei trapezi

Stimare la distanza percorsa da un veicolo utilizzando la velocit&#xE0; registrata in funzione del tempo &#xE8; un problema comune in fisica e in ingegneria. Quando i dati di velocit&#xE0; sono disponibili a intervalli di tempo discreti, anzich&#xE9; come una funzione continua, si ricorre spesso a metodi di integrazione numerica, come la regola dei trapezi, per approssimare lo spostamento totale.
La regola dei trapezi consiste nel suddividere l&#x2019;intervallo di tempo complessivo in pi&#xF9; sottointervalli di uguale ampiezza. In ciascun sottointervallo, le velocit&#xE0; registrate agli estremi vengono congiunte con un segmento di retta, formando un trapezio nel grafico velocit&#xE0;&#x2013;tempo. Ogni trapezio presenta due lati verticali che rappresentano le velocit&#xE0; all&#x2019;inizio e alla fine del sottointervallo e una base orizzontale corrispondente alla durata del sottointervallo.
Per approssimare lo spostamento, si calcola l&#x2019;area di ciascun trapezio. Tale area rappresenta la media delle velocit&#xE0; agli estremi del sottointervallo, moltiplicata per la durata del sottointervallo. Sommando le aree di tutti i trapezi si ottiene una stima della distanza percorsa complessivamente nell&#x2019;intero intervallo di tempo.
In questo procedimento, le velocit&#xE0; registrate all&#x2019;inizio e alla fine dell&#x2019;intervallo complessivo vengono utilizzate una sola volta, mentre le velocit&#xE0; misurate nei punti intermedi contribuiscono a due trapezi adiacenti. Questo approccio sistematico fornisce un&#x2019;approssimazione affidabile dello spostamento, soprattutto quando il numero di sottointervalli &#xE8; sufficientemente elevato. La regola dei trapezi &#xE8; quindi una tecnica fondamentale dell&#x2019;analisi numerica, poich&#xE9; offre un metodo pratico per integrare dati empirici relativi al moto o ad altri processi dipendenti dal tempo.

Regola del punto medio e regola dei trapezi

Stimare la distanza percorsa da un veicolo utilizzando la velocit&#xE0; registrata in funzione del tempo &#xE8; un problema comune in fisica e in ...

Education

Didattica

JoVE Core

Calculus

Analisi matematica

Unit

Unità

Techniques of Integration

Tecniche di integrazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

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integration by parts: definite integrals

Using Parts and Substitution for Area

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Solving Secant and Tangent Integrals

Integrals of secant and tangent powers are often solved by choosing a trig substitution that matches the integrand. The key idea is to use the parity of the exponents, which means whether a power is even or odd. A good substitution lets part of the integrand pair with the derivative of a trig function. When the power of secant is even, tangent is the better substitution variable. Since the derivative of tangent is secant squared, a factor of sec^2 x can be split off. The remaining even power...

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rationalizing substitutions

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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

Estimating Area with Simpson’s Rule

Simpson’s Rule is a numerical integration method for estimating the value of a definite integral. It is useful when an exact antiderivative is hard or impossible to find. The method works by fitting a unique parabola through three equally spaced points on a curve and then integrating that quadratic over the interval. This approach can give a very accurate area estimate from only a few sampled values. It works well for many smoothly varying functions. That makes it a practical tool when a curve...

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improper integrals: discontinuous integrands

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Improper Integrals at an Endpoint

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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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