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

Imagine um caminhão circulando por uma rodovia. Para encontrar a distância que ele percorre durante um dado intervalo de tempo t, encontre a área sob seu gráfico velocidade–tempo.Como a velocidade do caminhão é conhecida em pontos de tempo discretos, essa área não pode ser calculada com precisão. Em vez disso, a área é aproximada usando a regra trapezoidal. Para isso, divida o intervalo total de tempo em n subintervalos iguais, cada um com largura de Δt.Nesse método, os dois extremos de cada subintervalo são conectados por uma linha reta, formando trapézios.Para um dado subintervalo, que possui dois lados verticais representando as velocidades e uma base horizontal representando a largura do tempo, a área desse trapézio é a média dos comprimentos de seus dois lados verticais multiplicada pela distância horizontal.Somar as áreas de todos os n trapézios fornece uma aproximação do deslocamento.Após manipulações algébricas e a resolução dessa soma, a primeira e a última velocidades aparecem uma vez, enquanto as velocidades internas aparecem duas vezes porque cada uma pertence a dois trapézios.Após combinar termos semelhantes, a expressão resultante aproxima o deslocamento total ao longo do intervalo de tempo especificado.

regras do ponto médio e da regra do trapézio

Estimar a dist&#xE2;ncia percorrida por um ve&#xED;culo com a velocidade registrada ao longo do tempo &#xE9; um problema comum em f&#xED;sica e engenharia. Quando os dados de velocidade est&#xE3;o dispon&#xED;veis em intervalos de tempo discretos, em vez de como uma fun&#xE7;&#xE3;o cont&#xED;nua, m&#xE9;todos de integra&#xE7;&#xE3;o num&#xE9;rica, como a regra do trap&#xE9;zio, s&#xE3;o frequentemente empregados para aproximar o deslocamento total.
A regra do trap&#xE9;zio funciona dividindo o intervalo de tempo total em v&#xE1;rios segmentos iguais. Dentro de cada segmento, as velocidades registradas nos pontos extremos s&#xE3;o ligadas por uma linha reta, formando um trap&#xE9;zio em um gr&#xE1;fico de velocidade&#x2013;tempo. Cada trap&#xE9;zio tem um lado vertical representando a velocidade no in&#xED;cio e no fim do segmento e uma base horizontal correspondente ao intervalo de tempo.
Para aproximar o deslocamento, calcula-se a &#xE1;rea de cada trap&#xE9;zio. Essa &#xE1;rea representa a m&#xE9;dia das velocidades nos pontos extremos do segmento, multiplicada pela dura&#xE7;&#xE3;o do segmento. A soma das &#xE1;reas de todos os trap&#xE9;zios fornece uma estimativa do deslocamento total ao longo de todo o per&#xED;odo.
Nesse processo, as velocidades registradas no in&#xED;cio e no fim do intervalo de tempo total s&#xE3;o usadas apenas uma vez, enquanto as velocidades nos pontos intermedi&#xE1;rios entram no c&#xE1;lculo de dois trap&#xE9;zios adjacentes. Essa abordagem sistem&#xE1;tica resulta em uma aproxima&#xE7;&#xE3;o confi&#xE1;vel do deslocamento, especialmente quando o n&#xFA;mero de intervalos &#xE9; suficientemente grande. A regra do trap&#xE9;zio &#xE9;, portanto, uma t&#xE9;cnica fundamental em an&#xE1;lise num&#xE9;rica, fornecendo uma maneira pr&#xE1;tica de integrar dados emp&#xED;ricos de movimentos ou de outros processos dependentes do tempo.

Regras do ponto médio e da regra do trapézio

Estimar a dist&#xE2;ncia percorrida por um ve&#xED;culo com a velocidade registrada ao longo do tempo &#xE9; um problema comum em f&#xED;sica e ...

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Calculus

Cálculo

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Techniques of Integration

Técnicas de Integração

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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

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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.

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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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