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

Imagina un camión moviéndose por una autopista. Para encontrar la distancia que recorre durante un intervalo de tiempo dado t, se encuentra el área bajo su gráfica velocidad-tiempo.Dado que la velocidad del camión se conoce en puntos temporales discretos, esta área no puede calcularse con precisión. En su lugar, el área se aproxima usando la regla del trapecio. Para ello, divida el intervalo total de tiempo en n subintervalos iguales, cada uno con un ancho de Δt.En este método, los dos extremos de cada subintervalo están conectados por una línea recta, formando trapecios.Para un subintervalo dado, que tiene dos lados verticales que representan las velocidades y una base horizontal que representa el ancho temporal, el área de este trapecio es el promedio de las longitudes de sus dos lados verticales multiplicado por la distancia horizontal.Sumar las áreas de todos los n trapecioides da una aproximación del desplazamiento.Tras manipulaciones algebraicas y resolver esta suma, la primera y la última velocidades aparecen una vez, mientras que las velocidades interiores aparecen dos veces porque cada una pertenece a dos trapecioides.Tras combinar términos similares, la expresión resultante aproxima el desplazamiento total durante el intervalo de tiempo especificado.

reglas del punto medio y del trapecio

Estimar la distancia recorrida por un veh&#xED;culo utilizando su velocidad registrada a lo largo del tiempo es un problema com&#xFA;n en f&#xED;sica e ingenier&#xED;a. Cuando se dispone de datos de velocidad en intervalos de tiempo discretos, en lugar de como una funci&#xF3;n continua, se suelen emplear m&#xE9;todos de integraci&#xF3;n num&#xE9;rica como la regla del trapecio para aproximar el desplazamiento total.
La regla del trapecio funciona dividiendo el intervalo de tiempo total en varios segmentos iguales. Dentro de cada segmento, las velocidades registradas en los extremos se conectan con una l&#xED;nea recta, formando un trapecio en una gr&#xE1;fica de velocidad&#x2013;tiempo. Cada trapecio tiene un lado vertical que representa la velocidad en los extremos del segmento y una base horizontal que corresponde al intervalo de tiempo.
Para aproximar el desplazamiento, se calcula el &#xE1;rea de cada trapecio. Esta &#xE1;rea representa el promedio de las velocidades en los extremos del segmento, multiplicado por la duraci&#xF3;n del segmento. La suma de las &#xE1;reas de todos los trapecios proporciona una estimaci&#xF3;n del desplazamiento total recorrido durante todo el per&#xED;odo de tiempo.
En este proceso, las velocidades registradas en los extremos del intervalo de tiempo total se utilizan solo una vez, mientras que las velocidades en los puntos intermedios contribuyen a dos trapecios adyacentes. Este enfoque sistem&#xE1;tico proporciona una aproximaci&#xF3;n fiable del desplazamiento, especialmente cuando el n&#xFA;mero de intervalos es suficientemente grande. Por lo tanto, la regla del trapecio es una t&#xE9;cnica fundamental en el an&#xE1;lisis num&#xE9;rico, que proporciona una forma pr&#xE1;ctica de integrar datos emp&#xED;ricos sobre el movimiento u otros procesos dependientes del tiempo.

Reglas del punto medio y del trapecio

Estimar la distancia recorrida por un veh&#xED;culo utilizando su velocidad registrada a lo largo del tiempo es un problema com&#xFA;n en f&#xED;sica e ...

Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Techniques of Integration

Técnicas de integración

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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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integration by parts: definite integrals

Using Parts and Substitution for Area

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Solving Trig Integrals with Odd and Even Powers

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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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