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

El problema de la distancia determina hasta dónde ha viajado un objeto usando la velocidad medida en diferentes momentos del tiempo.Cuando la velocidad varía, la distancia total puede aproximarse añadiendo pequeños intervalos de desplazamiento, cada uno mostrando el movimiento en un corto paso de tiempo.Por ejemplo, en una carrera, un corredor acelera de forma constante durante los primeros tres segundos. Las mediciones de velocidad realizadas cada medio segundo muestran que la velocidad aumenta de 0 a 6,2 metros por segundo.Estas mediciones se utilizan para estimar la distancia total dividiendo el gráfico velocidad-tiempo en rectángulos de medio segundo para la suma inferior y la suma superior.La estimación inferior utiliza velocidades en los extremos izquierdos de cada intervalo de tiempo. Multiplicando cada una de estas velocidades por el paso de tiempo de medio segundo y sumando los resultados individuales, se obtienen 10,55 metros.Por otro lado, la estimación superior utiliza velocidades finales derechas. Multiplicando cada uno por el paso de tiempo y sumando se obtienen 13,65 metros. La distancia real se sitúa entre estas dos estimaciones. Aumentar el número de mediciones conduce a un resultado más preciso.Con mediciones infinitas, la distancia es igual al área bajo la curva, mostrada por la integral de la velocidad a lo largo del tiempo.

problema de distancia

Cuando la velocidad de un objeto cambia con el tiempo, la distancia total recorrida se puede determinar sumando peque&#xF1;os intervalos de desplazamiento durante pasos de tiempo cortos. Este enfoque aproxima la distancia total real mediante la suma num&#xE9;rica y el uso del c&#xE1;lculo integral. Se puede obtener una estimaci&#xF3;n del desplazamiento total midiendo la velocidad a intervalos regulares y multiplicando cada valor por el paso de tiempo correspondiente.Si un corredor acelera durante los primeros 3 s de una carrera, las mediciones de velocidad tomadas cada 0,5 s indican un aumento de 0 a 6,2 m/s. Un m&#xE9;todo de estimaci&#xF3;n utiliza las velocidades de los extremos izquierdos, donde cada valor de velocidad se multiplica por el paso de tiempo de 0,5 s, lo que produce una distancia aproximada de 10,55 m.Otro m&#xE9;todo utiliza las velocidades del extremo derecho, obteniendo una estimaci&#xF3;n ligeramente superior de 13,65 m. La distancia real recorrida se encuentra entre estas dos estimaciones, y las mediciones de velocidad m&#xE1;s frecuentes mejoran la precisi&#xF3;n.Una determinaci&#xF3;n m&#xE1;s precisa de la distancia se basa en el concepto de l&#xED;mites. El desplazamiento exacto se puede calcular tomando el l&#xED;mite de la suma de los valores de velocidad multiplicados por peque&#xF1;os pasos de tiempo &#x394;t a medida que aumenta el n&#xFA;mero de mediciones. Matem&#xE1;ticamente, esto se expresa como:\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;Este l&#xED;mite corresponde a la definici&#xF3;n de la suma de Riemann de la integral, lo que demuestra que la distancia total recorrida es la integral de la velocidad con respecto al tiempo. A medida que los pasos de tiempo se hacen infinitesimalmente peque&#xF1;os, la suma converge con precisi&#xF3;n a la integral, eliminando el error de aproximaci&#xF3;n asociado con los m&#xE9;todos de suma num&#xE9;rica.

Problema de distancia

Cuando la velocidad de un objeto cambia con el tiempo, la distancia total recorrida se puede determinar sumando peque&#xF1;os intervalos de desplazamiento ...

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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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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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