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

Lorsque deux variables ou plus sont liées par une équation, un changement dans l’une affecte les autres. Par exemple, dans un ballon en expansion, le volume et le rayon changent ensemble au fil du temps. En différenciant l’équation du volume par rapport au temps, on relie les taux auxquels les deux changent.Ce concept de « Taux Associés » est essentiel pour gonfler en toute sécurité une montgolfière. L’enveloppe du ballon gonflé est modélisée comme une sphère parfaite pour l’analyse mathématique.L’air est pompé dans le ballon à un débit constant qui entraîne un changement continu de volume au fil du temps.Supposons que le rayon soit de 2 mètres et que l’air entre à un rythme de 0,5 mètre cube par seconde. En substituant ces grandeurs connues dans l’équation du débit, on obtient la vitesse à laquelle le ballon se dilate à cet instant.De manière cruciale, le tissu a un taux d’expansion maximal. Si le rayon s’étend trop vite, le matériau se déchire aussi.Puisque la vitesse d’expansion est en réalité la plus élevée lorsque le ballon est petit, le calcul de ce taux garantit que l’inflation initiale rapide ne dépasse pas le seuil de rupture du matériau.

taux de variation liés

Related Rates

Lorsque deux ou plusieurs grandeurs physiques sont reli&#xE9;es par une seule relation, toute variation de l&#x2019;une entra&#xEE;ne n&#xE9;cessairement une modification des autres. Cette interd&#xE9;pendance constitue la base de l&#x2019;analyse des taux de variation li&#xE9;s, qui examine comment diff&#xE9;rentes grandeurs varient au cours du temps. Un exemple physique classique est celui d&#x2019;un ballon qui se gonfle, dont la taille varie continuellement &#xE0; mesure que de l&#x2019;air y est ajout&#xE9;.Dans le cas d&#x2019;une montgolfi&#xE8;re, l&#x2019;enveloppe gonfl&#xE9;e est souvent assimil&#xE9;e &#xE0; une sph&#xE8;re parfaite pour simplifier l&#x2019;analyse math&#xE9;matique. Selon cette hypoth&#xE8;se, le volume du ballon d&#xE9;pend directement de son rayon, conform&#xE9;ment &#xE0; la relation g&#xE9;om&#xE9;trique suivante<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>&#xC0; mesure que l&#x2019;air est insuffl&#xE9; dans le ballon, le volume augmente r&#xE9;guli&#xE8;rement dans le temps, et cette augmentation entra&#xEE;ne &#xE9;galement l&#x2019;augmentation du rayon. En d&#xE9;rivant cette relation par rapport au temps, on &#xE9;tablit un lien entre le d&#xE9;bit d&#x2019;air entrant dans le ballon et la vitesse de variation du rayon.Ce lien est essentiel pour garantir un gonflage en toute s&#xE9;curit&#xE9;. L&#x2019;air entre dans le ballon &#xE0; un d&#xE9;bit constant, ce qui entra&#xEE;ne un taux d&#x2019;augmentation du volume connu. &#xC0; un instant donn&#xE9;, lorsque le rayon est de 2 m et que le d&#xE9;bit d&#x2019;air entrant est de 0,5 m^3/s, ces valeurs peuvent &#xEA;tre utilis&#xE9;es pour d&#xE9;terminer la vitesse &#xE0; laquelle le rayon augmente &#xE0; cet instant pr&#xE9;cis. Cette vitesse correspond &#xE0; la rapidit&#xE9; avec laquelle l&#x2019;enveloppe du ballon est mise en tension.Les contraintes mat&#xE9;rielles rendent ce calcul essentiel. L&#x2019;enveloppe du ballon ne peut r&#xE9;sister qu&#x2019;&#xE0; une vitesse d&#x2019;expansion maximale avant qu&#x2019;une d&#xE9;chirure ne se produise. Il importe de souligner que la vitesse d&#x2019;expansion radiale est maximale lorsque le ballon est petit et diminue &#xE0; mesure que le ballon se gonfle. En &#xE9;valuant la vitesse d&#x2019;expansion durant les premi&#xE8;res &#xE9;tapes du gonflage, les op&#xE9;rateurs peuvent s&#x2019;assurer que le gonflage initial rapide ne d&#xE9;passe pas le seuil de rupture du mat&#xE9;riau, afin de pr&#xE9;server l&#x2019;int&#xE9;grit&#xE9; structurelle et la s&#xE9;curit&#xE9;.

Taux de variation liés

Lorsque deux ou plusieurs grandeurs physiques sont reli&#xE9;es par une seule relation, toute variation de l&#x2019;une entra&#xEE;ne n&#xE9;cessairement ...

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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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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...

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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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