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

Il problema della distanza determina quanto lontano ha viaggiato un oggetto usando la velocità misurata in diversi momenti temporali.Quando la velocità varia, la distanza totale può essere approssimata aggiungendo piccoli intervalli di spostamento, ognuno dei quali mostra il moto su un breve intervallo di tempo.Ad esempio, in una gara, un corridore accelera costantemente durante i primi tre secondi. Le misurazioni della velocità effettuate ogni mezzo secondo mostrano un aumento della velocità da 0 a 6,2 metri al secondo.Queste misurazioni vengono utilizzate per stimare la distanza totale dividendo il grafico velocità–tempo in rettangoli di mezzo secondo per la somma inferiore e quella superiore.La stima inferiore utilizza le velocità agli estremi sinistri di ogni intervallo di tempo. Moltiplicando ciascuna di queste velocità per il passo temporale di mezzo secondo e sommando i singoli risultati, si ottengono 10,55 metri.D'altra parte, la stima superiore utilizza le velocità di estremità destra. Moltiplicando ciascuno di questi per il passo temporale e sommando si ottengono 13,65 metri. La distanza effettiva si colloca tra queste due stime. Aumentare il numero di misurazioni porta a un risultato più accurato.Con misurazioni infinite, la distanza è uguale all'area sotto la curva, mostrata dall'integrale della velocità nel tempo.

problema della distanza

Quando la velocit&#xE0; di un oggetto varia nel tempo, la distanza totale percorsa pu&#xF2; essere determinata sommando piccoli intervalli di spostamento corrispondenti a brevi incrementi di tempo. Questo approccio approssima la distanza reale mediante una sommatoria numerica e il ricorso al calcolo integrale. Una stima dello spostamento totale pu&#xF2; essere ottenuta misurando la velocit&#xE0; a intervalli regolari e moltiplicando ciascun valore per il corrispondente passo temporale.Se un corridore accelera nei primi tre secondi di una gara, misurazioni della velocit&#xE0; effettuate ogni mezzo secondo indicano un aumento da 0 a 6,2 m/s. Un metodo di stima consiste nell&#x2019;utilizzare i valori di velocit&#xE0; all&#x2019;estremit&#xE0; sinistra di ciascun intervallo, per cui ciascun valore viene moltiplicato per il passo temporale di 0,5 s, ottenendo una distanza approssimata di 10,55 m.Un altro metodo applica i valori di velocit&#xE0; all&#x2019;estremit&#xE0; destra, fornendo una stima leggermente pi&#xF9; elevata di 13,65 m. La distanza effettivamente percorsa si colloca tra queste due stime, mentre misurazioni della velocit&#xE0; pi&#xF9; frequenti migliorano la precisione.Una determinazione pi&#xF9; precisa della distanza si basa sul concetto di limite. Lo spostamento esatto si ottiene considerando il limite della somma dei valori di velocit&#xE0; moltiplicati per piccoli intervalli di tempo quando aumenta il numero di misurazioni. In termini matematici, ci&#xF2; si esprime come segue:\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;Questo limite corrisponde alla definizione dell&#x2019;integrale come limite di somme di Riemann, dimostrando che la distanza totale percorsa &#xE8; l&#x2019;integrale della velocit&#xE0; rispetto al tempo. Quando gli intervalli di tempo diventano infinitesimali, la somma converge esattamente all&#x2019;integrale, eliminando l&#x2019;errore di approssimazione associato ai metodi di sommatoria numerica.

Problema della distanza

Quando la velocit&#xE0; di un oggetto varia nel tempo, la distanza totale percorsa pu&#xF2; essere determinata sommando piccoli intervalli di spostamento ...

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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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key terms and concepts

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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