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

O problema da distância determina quão longe um objeto percorreu usando a velocidade medida em diferentes momentos do tempo.Quando a velocidade varia, a distância total pode ser aproximada adicionando pequenos intervalos de deslocamento, cada um mostrando o movimento em um curto intervalo de tempo.Por exemplo, em uma corrida, um corredor acelera de forma constante durante os primeiros três segundos. Medições de velocidade feitas a cada meio segundo mostram que a velocidade aumenta de 0 para 6,2 metros por segundo.Essas medições são usadas para estimar a distância total dividindo o gráfico velocidade–tempo em retângulos de meio segundo para a soma inferior e a soma superior.A estimativa inferior usa velocidades nos extremos esquerdos de cada intervalo de tempo. Multiplicando cada uma dessas velocidades pelo passo de tempo de meio segundo e somando os resultados individuais, obtém-se 10,55 metros.Por outro lado, a estimativa superior usa velocidades de extremidade à direita. Multiplicando cada um deles pelo passo de tempo e somando, obtém-se 13,65 metros. A distância real está entre essas duas estimativas. Aumentar o número de medições leva a um resultado mais preciso.Com medições infinitas, a distância é igual à área sob a curva, mostrada pela integral da velocidade ao longo do tempo.

problema da distância

Quando a velocidade de um objeto varia ao longo do tempo, a dist&#xE2;ncia total percorrida pode ser determinada pela soma de pequenos intervalos de deslocamento em intervalos de tempo curtos. Essa abordagem aproxima a dist&#xE2;ncia real por meio de somas num&#xE9;ricas e do uso do c&#xE1;lculo integral. Uma estimativa do deslocamento total pode ser obtida medindo a velocidade em intervalos regulares e multiplicando-se cada valor pelo intervalo de tempo correspondente.Se um corredor acelera durante os tr&#xEA;s primeiros segundos de uma corrida, as medi&#xE7;&#xF5;es de velocidade feitas a cada meio segundo indicam um aumento de 0 a 6,2 m/s. Um m&#xE9;todo de estimativa envolve o uso das velocidades no ponto inicial de cada intervalo, onde cada valor de velocidade &#xE9; multiplicado pelo intervalo de tempo de 0,5 s, o que resulta em uma dist&#xE2;ncia aproximada de 10,55 m.Outro m&#xE9;todo aplica as velocidades no extremo direito do intervalo, resultando em uma estimativa ligeiramente maior de 13,65 m. A dist&#xE2;ncia real percorrida situa-se entre essas duas estimativas, sendo que medi&#xE7;&#xF5;es de velocidade mais frequentes melhoram a precis&#xE3;o.Uma determina&#xE7;&#xE3;o mais precisa da dist&#xE2;ncia baseia-se no conceito de limites. O deslocamento exato pode ser encontrado ao se tomar o limite da soma dos valores de velocidade multiplicados por pequenos intervalos de tempo, &#xE0; medida que o n&#xFA;mero de medi&#xE7;&#xF5;es aumenta. Matematicamente, isso &#xE9; expresso 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;Esse limite corresponde &#xE0; defini&#xE7;&#xE3;o da integral pela soma de Riemann, demonstrando que a dist&#xE2;ncia total percorrida &#xE9; a integral da velocidade em fun&#xE7;&#xE3;o do tempo. &#xC0; medida que os intervalos de tempo se tornam infinitesimalmente pequenos, a soma converge exatamente para a integral, eliminando o erro de aproxima&#xE7;&#xE3;o associado aos m&#xE9;todos de soma num&#xE9;rica.

Problema da Distância

Quando a velocidade de um objeto varia ao longo do tempo, a dist&#xE2;ncia total percorrida pode ser determinada pela soma de pequenos intervalos de ...

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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.

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Estimating Distance from Changing Velocity

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