Name: balloon inflation and changing rates
Uploaded: 2026-01-12T12:24:18.000Z
Duration: 1 min 18 s
Channel: JoVE

Quando due o più variabili sono collegate da un'equazione, un cambiamento in una influenza le altre. Ad esempio, in un palloncino in espansione, volume e raggio cambiano insieme nel tempo. Derivare l'equazione del volume rispetto al tempo collega le velocità a cui entrambe cambiano.Questo concetto di 'Tasse Correlate' è fondamentale per gonfiare in sicurezza una mongolfiera. L'inviluppo del palloncino gonfiato è modellata come una sfera perfetta per l'analisi matematica.L'aria viene pompata nel pallone a una velocità costante che favorisce una variazione continua di volume nel tempo.Supponiamo che il raggio sia di 2 metri e che l'aria entri a una velocità di 0,5 metri cubi al secondo. Sostituendo queste quantità note nell'equazione del tasso, si ottiene la velocità con cui il palloncino si sta espandendo in questo istante.Fondamentale, il tessuto ha un tasso massimo di espansione. Se il raggio si espande troppo rapidamente, il materiale si strappa.Poiché la velocità di espansione è in realtà più elevata quando il palloncino è piccolo, calcolare questa velocità garantisce che il rapido gonfiaggio iniziale non superi la soglia di rottura del materiale.

tassi di variazione correlati

Quando due o pi&#xF9; grandezze fisiche sono collegate da un&#x2019;unica relazione, una variazione di una di esse influisce necessariamente sulle altre. Questa interdipendenza costituisce la base dell&#x2019;analisi dei tassi di variazione correlati, che esamina come diverse grandezze varino nel tempo. Un classico esempio fisico &#xE8; un pallone in espansione, le cui dimensioni cambiano continuamente man mano che si aggiunge aria.Nel caso di una mongolfiera, l&#x2019;involucro gonfiato &#xE8; comunemente idealizzato come una sfera perfetta per semplificare l&#x2019;analisi matematica. In base a questa ipotesi, il volume della mongolfiera dipende direttamente dal suo raggio secondo la relazione geometrica:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19808/19808_Equation_1_editor_19808.svg" alt="19808_Equation_1.svg" /></figure>Man mano che l&#x2019;aria viene pompata nella mongolfiera, il volume aumenta costantemente nel tempo e tale aumento costringe anche il raggio ad aumentare. Differenziando questa relazione rispetto al tempo, si collega la portata con cui l&#x2019;aria entra nella mongolfiera al tasso di variazione del raggio.Questa connessione &#xE8; essenziale per garantire un gonfiaggio sicuro. L&#x2019;aria entra nella mongolfiera a una portata costante, determinando un tasso noto di aumento del volume. In un istante specifico, ad esempio quando il raggio &#xE8; 2 m e la portata di ingresso &#xE8; 0,5 m^3/s, questi valori noti possono essere utilizzati per determinare quanto rapidamente il raggio stia aumentando in quel momento. Tale tasso rappresenta la rapidit&#xE0; con cui il tessuto dell&#x2019;involucro viene stirato.Le limitazioni dei materiali rendono questo calcolo cruciale. L&#x2019;involucro della mongolfiera pu&#xF2; sopportare solo un tasso massimo di espansione prima che si verifichi una lacerazione. &#xC8; importante notare che il tasso di espansione radiale &#xE8; massimo quando la mongolfiera &#xE8; piccola e diminuisce man mano che cresce. Valutando il tasso di espansione nelle prime fasi del gonfiaggio, gli operatori possono assicurarsi che la rapida crescita iniziale non superi la soglia di rottura del materiale, contribuendo a mantenere l&#x2019;integrit&#xE0; strutturale e la sicurezza.

Tassi di variazione correlati

Quando due o pi&#xF9; grandezze fisiche sono collegate da un&#x2019;unica relazione, una variazione di una di esse influisce necessariamente sulle altre. ...

Education

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JoVE Core

Calculus

Analisi matematica

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Unità

Differentiation Rules

Regole di derivazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

Related Rates

balloon inflation and changing rates

Balloon inflation is a clear example of related rates, where one changing quantity affects another through a single relationship. Related rates study how linked quantities change with respect to time. As air is added, the balloon’s volume rises, and the radius must also increase.

For a hot air balloon, the inflated envelope is often modeled as a perfect sphere. This simplification makes the math easier to work with. In that model, the balloon’s volume depends directly on its radius.

When air is pumped in at a constant rate, the volume increases at a known speed. Differentiating the volume-radius relationship with respect to time connects that inflow rate to the rate of change of the radius. If the radius is 2 meters and the inflow rate is 0.5 cubic meters per second, those values can be used to find how fast the radius is expanding at that instant.

That radial growth shows how quickly the balloon fabric is being stretched. This calculation matters because the envelope can only handle a limited expansion rate before tearing. The rate of radial expansion is greatest when the balloon is small, and it decreases as the balloon grows. By checking the early stages of inflation, operators can keep the expansion below the rupture threshold and help maintain safety and structural integrity.

Balloon Inflation and Changing Rates

derivatives of simple functions

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Derivatives show how fast a function changes, and they also describe slope. A derivative can tell the slope of a straight line or the slope of a tangent line to a curve at one point. In this lesson, the idea is tied to a roller coaster track, where the derivative of the horizontal position tells how steep the ride is at each spot. A flat roller coaster section can be modeled with a constant function, f(x) = c, where c is a fixed height. Its derivative is f'(x) = 0, so the slope is zero and...

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limits on trigonometric functions

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the chain rule

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implicit differentiation

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e from Natural Log Derivatives

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derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

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application of rates of change

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exponential growth

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related rates

Balloon inflation is a clear example of related rates, where one changing quantity affects another through a single relationship. Related rates study how linked quantities change with respect to time. As air is added, the balloon’s volume rises, and the radius must also increase. For a hot air balloon, the inflated envelope is often modeled as a perfect sphere. This simplification makes the math easier to work with. In that model, the balloon’s volume depends directly on its radius. When air...

linearization and approximation

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application of linearization and approximation

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Estimating Altitude Changes with Linearization

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hyperbolic functions

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inverse hyperbolic functions and their derivatives

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key terms and concepts

Learn how related rates explain balloon inflation, connecting air flow, volume, and radius changes to estimate expansion speed and help prevent tearing.

Balloon Inflation and Changing Rates | 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

Balloon Inflation and Changing Rates

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