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

Problem odległości określa, jak daleko obiekt przeszedł, używając prędkości mierzonej w różnych momentach czasu.Gdy prędkość się zmienia, całkowitą odległość można przybliżyć, dodając krótkie przedziały przemieszczeń, z których każdy pokazuje ruch w krótkim kroku czasowym.Na przykład w wyścigu biegacz przyspiesza systematycznie przez pierwsze trzy sekundy. Pomiary prędkości wykonywane co pół sekundy pokazują, że prędkość wzrasta od 0 do 6,2 metra na sekundę.Pomiary te służą do oszacowania całkowitej odległości poprzez podział wykresu prędkości i czasu na półsekundowe prostokąty dla sumy dolnej i górnej.Dolne oszacowanie wykorzystuje prędkości na lewych końcach każdego przedziału czasowego. Mnożenie każdej z tych prędkości przez krok czasowy pół sekundy i sumowanie poszczególnych wyników daje 10,55 metra.Z kolei górne oszacowanie wykorzystuje prędkości prawej końcowej. Mnożenie każdego z nich przez krok czasowy i dodawanie daje 13,65 metra. Rzeczywista odległość leży między tymi dwoma szacunkami. Zwiększenie liczby pomiarów prowadzi do dokładniejszego wyniku.W przypadku pomiarów nieskończonych odległość jest równa powierzchni pod krzywą, co jest pokazane przez całkę prędkości w czasie.

problem wyznaczania drogi

Gdy pr&#x119;dko&#x15B;&#x107; obiektu zmienia si&#x119; w czasie, ca&#x142;kowit&#x105; odleg&#x142;o&#x15B;&#x107; przebyt&#x105; mo&#x17C;na okre&#x15B;li&#x107;, sumuj&#x105;c ma&#x142;e przyrosty przemieszczenia w kr&#xF3;tkich odst&#x119;pach czasu. To podej&#x15B;cie przybli&#x17C;a rzeczywist&#x105; odleg&#x142;o&#x15B;&#x107; poprzez sumowanie numeryczne i wykorzystanie rachunku ca&#x142;kowego. Oszacowanie ca&#x142;kowitego przemieszczenia mo&#x17C;na uzyska&#x107;, mierz&#x105;c pr&#x119;dko&#x15B;&#x107; w regularnych odst&#x119;pach czasu i mno&#x17C;&#x105;c ka&#x17C;d&#x105; warto&#x15B;&#x107; przez odpowiadaj&#x105;cy jej przedzia&#x142; czasowy.Je&#x15B;li biegacz przyspiesza w ci&#x105;gu pierwszych 3 s biegu, pomiary pr&#x119;dko&#x15B;ci wykonywane co p&#xF3;&#x142; sekundy wskazuj&#x105; na wzrost od 0 do 6,2 m/s. Jedna z metod szacowania polega na u&#x17C;yciu pr&#x119;dko&#x15B;ci w lewych punktach ko&#x144;cowych, gdzie ka&#x17C;da warto&#x15B;&#x107; pr&#x119;dko&#x15B;ci jest mno&#x17C;ona przez przedzia&#x142; czasowy 0,5 s, co daje przybli&#x17C;on&#x105; odleg&#x142;o&#x15B;&#x107; wynosz&#x105;c&#x105; 10,55 m.Inna metoda wykorzystuje pr&#x119;dko&#x15B;ci w prawych punktach ko&#x144;cowych, daj&#x105;c nieco wy&#x17C;sz&#x105; warto&#x15B;&#x107; szacunkow&#x105;, wynosz&#x105;c&#x105; 13,65 m. Rzeczywista przebyta odleg&#x142;o&#x15B;&#x107; le&#x17C;y pomi&#x119;dzy tymi dwoma szacunkami, a cz&#x119;stsze pomiary pr&#x119;dko&#x15B;ci zwi&#x119;kszaj&#x105; dok&#x142;adno&#x15B;&#x107; oszacowania.Dok&#x142;adniejsze okre&#x15B;lenie odleg&#x142;o&#x15B;ci opiera si&#x119; na poj&#x119;ciu granicy. Dok&#x142;adne przemieszczenie mo&#x17C;na wyznaczy&#x107;, bior&#x105;c granic&#x119; sumy warto&#x15B;ci pr&#x119;dko&#x15B;ci pomno&#x17C;onych przez ma&#x142;e przyrosty czasu w miar&#x119; wzrostu liczby pomiar&#xF3;w. Matematycznie wyra&#x17C;a si&#x119; to nast&#x119;puj&#x105;co:\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;Granica ta odpowiada definicji ca&#x142;ki w postaci sumy Riemanna, kt&#xF3;ra pokazuje, &#x17C;e ca&#x142;kowita odleg&#x142;o&#x15B;&#x107; przebyta jest ca&#x142;k&#x105; pr&#x119;dko&#x15B;ci po czasie. W miar&#x119; jak przyrosty czasu staj&#x105; si&#x119; niesko&#x144;czenie ma&#x142;e, sumowanie zbiega si&#x119; dok&#x142;adnie do ca&#x142;ki, eliminuj&#x105;c b&#x142;&#x105;d aproksymacji zwi&#x105;zany z numerycznymi metodami sumowania.

Problem wyznaczania drogi

Gdy pr&#x119;dko&#x15B;&#x107; obiektu zmienia si&#x119; w czasie, ca&#x142;kowit&#x105; odleg&#x142;o&#x15B;&#x107; przebyt&#x105; mo&#x17C;na ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Integrals

Całki

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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distance problem

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...

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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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