Chute verticale d’un solide

I.   Chute verticale dans un fluide

1.  Forces exercées sur le solide

Soit un solide S en mouvement de chute verticale à proximité de la Terre. Ce solide est soumis à trois forces :

a.  Force exercée par la Terre : force de pesanteur

Un objet situé au voisinage de la Terre subit la force de gravitation F MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaamaaFiaabaGaamOraaGaay51Gaaaaa@3A4C@  qui peut s’identifier à la force de pesanteur P MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaamaaFiaabaGaamiuaaGaay51Gaaaaa@3A56@ .

Définition : On dit que la Terre crée un champ de pesanteur g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaamaaFiaabaGaam4zaaGaay51Gaaaaa@3A6D@ . Un objet de masse m MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaaiaad2gaaaa@38BF@  placé dans le champ de pesanteur g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaamaaFiaabaGaam4zaaGaay51Gaaaaa@3A6D@  subit une force P =m g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaamaaFiaabaGaamiuaaGaay51GaGaeyypa0JaamyBamaaFiaabaGaam4zaaGaay51Gaaaaa@3EEE@ .

Remarque : Le champ de pesanteur est supposé uniforme ( g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaamaaFiaabaGaam4zaaGaay51Gaaaaa@3A6D@  est le même en tout point) dans une zone pas trop étendue au voisinage de la Terre (en réalité, g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaaiaadEgaaaa@38B9@  diminue avec l’altitude).

b.  Force exercée par le fluide : poussée d’Archimède

Définition : Un corps totalement ou partiellement immergé dans un fluide subit une force F MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaamaaFiaabaGaamOraaGaay51Gaaaaa@3A4C@  verticale, de bas en haut, de valeur égale au poids du fluide déplacé appelée poussée d’Archimède.

F=ρVg MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaaiaadAeacqGH9aqpcqaHbpGCcaWGwbGaam4zaaaa@3D25@

avec{ F:poussée d'Archimède (N) ρ:masse volumique du fluide (kg. m -3 ) V: volume de la partie immergé du solide( m 3 ) g: valeur du champ de pesanteur MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefeKCPfgBaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biFbsj0=yi0dXdbba9pGe9xq=JbbG8vqFr0=vr0=vr0db8meaacaGaamaadaWaaiaacaabauaagaaakeaaiiaacaWFHbGaa8NDaiaa=vgacaWFJbGaaGPaVpaaceaabaqbaeaabqqaaaaabaGaa8Nraiaa=PdacaWFWbGaa83Baiaa=vhacaWFZbGaa83Caiaa=LoacaWFLbGaa8hiaiaa=rgacaWFNaGaa8xqaiaa=jhacaWFJbGaa8hAaiaa=LgacaWFTbGaa8h6aiaa=rgacaWFLbGaa8hiaiaa=HcacaWFobGaa8xkaaqaaiabeg8aYjaa=PdacaWFTbGaa8xyaiaa=nhacaWFZbGaa8xzaiaa=bcacaWF2bGaa83Baiaa=XgacaWF1bGaa8xBaiaa=LgacaWFXbGaa8xDaiaa=vgacaWFGaGaa8hzaiaa=vhacaWFGaGaa8Nzaiaa=XgacaWF1bGaa8xAaiaa=rgacaWFLbGaa8hiaiaa=HcacaWFRbGaa83zaiaa=5cacaWFTbWaaWbaaSqabeaacaWFTaGaa83maaaakiaa=LcaaeaacaWFwbGaa8Noaiaa=bcacaWF2bGaa83Baiaa=XgacaWF1bGaa8xBaiaa=vgacaWFGaGaa8hzaiaa=vgacaWFGaGaa8hBaiaa=fgacaWFGaGaa8hCaiaa=fgacaWFYbGaa8hDaiaa=LgacaWFLbGaa8hiaiaa=LgacaWFTbGaa8xBaiaa=vgacaWFYbGaa83zaiaa=LoacaWFGaGaa8hzaiaa=vhacaWFGaGaa83Caiaa=9gacaWFSbGaa8xAaiaa=rgacaWFLbGaa8hkaiaa=1gadaahaaWcbeqaaiaa=ndaaaGccaWFPaaabaGaa83zaiaa=PdacaWFGaGaa8NDaiaa=fgacaWFSbGaa8xzaiaa=vhacaWFYbGaa8hiaiaa=rgacaWF1bGaa8hiaiaa=ngacaWFObGaa8xyaiaa=1gacaWFWbGaa8hiaiaa=rgacaWFLbGaa8hiaiaa=bhacaWFLbGaa83Caiaa=fgacaWFUbGaa8hDaiaa=vgacaWF1bGaa8NCaaaaaiaawUhaaiaaykW7aaa@BA11@  

c.  Force de frottement exercée par le fluide

Soit un solide de vitesse v MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefeKCPfgBaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biFbsj0=yi0dXdbba9pGe9xq=JbbG8vqFr0=vr0=vr0db8meaacaGaamaadaWaaiaacaabauaagaaakeaadaWhcaqaaiaadAhaaiaawEniaaaa@3DD5@ . Le fluide exerce sur ce solide une force de frottement. Dans le cas d’une chute verticale dans un fluide, la force de frottement est de la forme f =k v MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefeKCPfgBaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biFbsj0=yi0dXdbba9pGe9xq=JbbG8vqFr0=vr0=vr0db8meaacaGaamaadaWaaiaacaabauaagaaakeaadaWhcaqaaiaadAgaaiaawEniaiabg2da9iabgkHiTiaadUgadaWhcaqaaiaadAhaaiaawEniaaaa@4357@ .

Remarque : La force f MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefeKCPfgBaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biFbsj0=yi0dXdbba9pGe9xq=JbbG8vqFr0=vr0=vr0db8meaacaGaamaadaWaaiaacaabauaagaaakeaadaWhcaqaaiaadAgaaiaawEniaaaa@3DC5@  est colinéaire au vecteur vitesse v MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefeKCPfgBaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biFbsj0=yi0dXdbba9pGe9xq=JbbG8vqFr0=vr0=vr0db8meaacaGaamaadaWaaiaacaabauaagaaakeaadaWhcaqaaiaadAhaaiaawEniaaaa@3DD5@  mais de sens opposé. Sa valeur est proportionnelle à la vitesse v.

Remarque : Pour des vitesses plus importantes, f peut être de la forme f=k v 2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefeKCPfgBaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biFbsj0=yi0dXdbba9pGe9xq=JbbG8vqFr0=vr0=vr0db8meaacaGaamaadaWaaiaacaabauaagaaakeaacaWGMbGaeyypa0Jaam4AaiaadAhadaahaaWcbeqaaiaaikdaaaaaaa@3FEB@ .

2.  Modélisation du mouvement

a.  Equation différentielle

Soit une bille de masse m et de vitesse initiale v 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaWaa8HaaeaacaWG2bWaaSbaaSqaaiaaicdaaeqaaaGccaGLxdcaaaa@3BE3@  en chute verticale dans un liquide.

La bille est soumise aux trois forces précédentes  (voir schéma ci-contre).

Référentiel utilisé : terrestre (galiléen par approximation)

2ème loi de Newton : F ext =m a G MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaWaaabqaeaadaWhcaqaaiaadAeadaWgaaWcbaGaamyzaiaadIhacaWG0baabeaaaOGaay51GaGaeyypa0JaamyBamaaFiaabaGaamyyamaaBaaaleaacaWGhbaabeaaaOGaay51GaaaleqabeqdcqGHris5aOGaaGPaVdaa@4717@   P + F + f =m a G MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaeyO0H4TaaGPaVlaaykW7daWhcaqaaiaadcfaaiaawEniaiabgUcaRmaaFiaabaGaamOraaGaay51GaGaey4kaSYaa8HaaeaacaWGMbaacaGLxdcacqGH9aqpcaWGTbWaa8HaaeaacaWGHbWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcaaaa@4CB7@

Sur l’axe ox : PFf=m a G MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiuaiabgkHiTiaadAeacqGHsislcaWGMbGaeyypa0JaamyBaiaadggadaWgaaWcbaGaam4raaqabaaaaa@4080@

mgρVgk v x =m a x (mρV)gk v x =m d v x dt d v x dt = k m v x + (mρV)g m MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@75FE@

Remarque : Cette équation différentielle est de la forme d v x dt =a v x +b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaamaalaaabaGaamizaiaadAhadaWgaaWcbaGaamiEaaqabaaakeaacaWGKbGaamiDaaaacqGH9aqpcaWGHbGaamODamaaBaaaleaacaWG4baabeaakiabgUcaRiaadkgaaaa@40BC@ .

b.  Etude expérimentale (voir TP)

La courbe v G =f(t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWGhbaabeaakiabg2da9iaadAgacaGGOaGaamiDaiaacMcaaaa@3E84@  a l’allure suivante. Elle possède deux régimes successifs :

c.  Détermination de la vitesse limite

Lorsque v G MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYhNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWGhbaabeaaaaa@3A48@  atteint la vitesse limite, on peut écrire : v G =cte MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWGhbaabeaakiabg2da9iaadogacaWG0bGaamyzaiaaykW7aaa@3F9E@

d v x dt =0 k m v lim + (mρV)g m =0 v lim = (mρV)g k MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOabaeqabaGaeyO0H4TaaGPaVlaaykW7daWcaaqaaiaadsgacaWG2bWaaSbaaSqaaiaadIhaaeqaaaGcbaGaamizaiaadshaaaGaeyypa0JaaGimaaqaaiabgkDiElaaykW7caaMc8+aaSaaaeaacqGHsislcaWGRbaabaGaamyBaaaacaWG2bWaaSbaaSqaaiGacYgacaGGPbGaaiyBaaqabaGccqGHRaWkdaWcaaqaaiaacIcacaWGTbGaeyOeI0IaeqyWdiNaamOvaiaacMcacaWGNbaabaGaamyBaaaacqGH9aqpcaaIWaaabaGaeyO0H4TaaGPaVlaaykW7caWG2bWaaSbaaSqaaiGacYgacaGGPbGaaiyBaaqabaGccqGH9aqpdaWcaaqaaiaacIcacaWGTbGaeyOeI0IaeqyWdiNaamOvaiaacMcacaWGNbaabaGaam4Aaaaaaaaa@6B04@

d.  Résolution de l’équation différentielle par une méthode numérique (méthode d’Euler)

On a d v x dt =a v x +b MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaamaalaaabaGaamizaiaadAhadaWgaaWcbaGaamiEaaqabaaakeaacaWGKbGaamiDaaaacqGH9aqpcaWGHbGaamODamaaBaaaleaacaWG4baabeaakiabgUcaRiaadkgaaaa@40BD@  avec a= k m MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadggacqGH9aqpdaWcaaqaaiabgkHiTiaadUgaaeaacaWGTbaaaaaa@3A9C@  et b= (mρV)g k MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadkgacqGH9aqpdaWcaaqaaiaacIcacaWGTbGaeyOeI0IaeqyWdiNaamOvaiaacMcacaWGNbaabaGaam4Aaaaaaaa@3F7D@ .

Si δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYhNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaeqiTdqMaamiDaaaa@3AF3@  est petit, on peut écrire a v x +b= δ v x δt δ v x =(a v x +b).δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadggacaWG2bWaaSbaaSqaaiaadIhaaeqaaOGaey4kaSIaamOyaiabg2da9maalaaabaGaeqiTdqMaamODamaaBaaaleaacaWG4baabeaaaOqaaiabes7aKjaadshaaaGaeyO0H4TaeqiTdqMaamODamaaBaaaleaacaWG4baabeaakiabg2da9iaacIcacaWGHbGaamODamaaBaaaleaacaWG4baabeaakiabgUcaRiaadkgacaGGPaGaaiOlaiabes7aKjaadshaaaa@52F1@

Supposons qu’à l’instant t=0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiDaiabg2da9iaaicdaaaa@3AFE@ , la vitesse du solide soit v x = v 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4baabeaakiabg2da9iaadAhadaWgaaWcbaGaaGimaaqabaaaaa@3D59@  (conditions initiales du mouvement).

à l’instant t=0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiDaiabg2da9iaaicdaaaa@3AFE@  :

v x = v 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4baabeaakiabg2da9iaadAhadaWgaaWcbaGaaGimaaqabaaaaa@3D59@

 

 

à l’instant t 1 =δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiDamaaBaaaleaacaaIXaaabeaakiabg2da9iabes7aKjaadshaaaa@3DD3@  :

v 1 = v 0 +δ v 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiabg2da9iaadAhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaH0oazcaWG2bWaaSbaaSqaaiaaicdaaeqaaaaa@4189@

=>

v 1 = v 0 +(a v 0 +b)δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiabg2da9iaadAhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaGGOaGaamyyaiaadAhadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGIbGaaiykaiabes7aKjaadshaaaa@4695@

à l’instant t 2 = t 1 +δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiDamaaBaaaleaacaaIYaaabeaakiabg2da9iaadshadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaH0oazcaWG0baaaa@40A0@  :

v 2 = v 1 +δ v 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaaIYaaabeaakiabg2da9iaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaH0oazcaWG2bWaaSbaaSqaaiaaigdaaeqaaaaa@418C@

=>

v 2 = v 1 +(a v 1 +b)δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaaIYaaabeaakiabg2da9iaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaGGOaGaamyyaiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGIbGaaiykaiabes7aKjaadshaaaa@4698@

etc...

  

 

 

Il est ainsi possible de calculer la vitesse v x MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4baabeaaaaa@3A69@  à chaque instant. La courbe v x =f(t) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4baabeaakiabg2da9iaadAgacaGGOaGaamiDaiaacMcaaaa@3EB6@  se rapproche de la courbe expérimentale si δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaeqiTdqMaamiDaaaa@3AE3@  est petit.

II.  Chute libre

1.  Définition

Un solide est dit en chute libre s’il est soumis uniquement à son poids (le fait qu’il n’existe pas de force de frottement impose que cette condition ne peut être réalisée que dans le vide).

Remarque : Lorsque la force de frottement du fluide et la poussée d’Archimède sont négligeables devant le poids, on peut considérer le solide comme étant en chute libre.

2.  Modélisation du mouvement

a.  Equation différentielle du mouvement

Système étudié : le solide.

Référentiel utilisé : terrestre (galiléen par approximation)

Bilan des forces extérieures : P MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaWaa8HaaeaacaWGqbaacaGLxdcaaaa@3ACE@  : poids du solide.

Deuxième loi de Newton : F ext =m a G MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaWaaabqaeaadaWhcaqaaiaadAeadaWgaaWcbaGaamyzaiaadIhacaWG0baabeaaaOGaay51GaGaeyypa0JaamyBamaaFiaabaGaamyyamaaBaaaleaacaWGhbaabeaaaOGaay51GaaaleqabeqdcqGHris5aOGaaGPaVdaa@4717@

P =m a G MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaaGPaVlabgkDiElaaykW7caaMc8+aa8HaaeaacaWGqbaacaGLxdcacqGH9aqpcaWGTbWaa8HaaeaacaWGHbWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcaaaa@475F@
m g =m a G MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaaGPaVlabgkDiElaaykW7caaMc8UaamyBamaaFiaabaGaam4zaaGaay51GaGaeyypa0JaamyBamaaFiaabaGaamyyamaaBaaaleaacaWGhbaabeaaaOGaay51Gaaaaa@4869@
a G = g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaaGPaVlabgkDiElaaykW7caaMc8+aa8HaaeaacaWGHbWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcacqGH9aqpdaWhcaqaaiaadEgaaiaawEniaaaa@4684@

Le vecteur accélération du centre d’inertie d’un solide en chute libre est égal au vecteur champ de pesanteur g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaWaa8HaaeaacaWGNbaacaGLxdcaaaa@3AE5@ .

La projection sur l’axe ox donne : a x =g d v x dt =g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadggadaWgaaWcbaGaamiEaaqabaGccqGH9aqpcaWGNbGaaGPaVlaaykW7cqGHshI3caaMc8UaaGPaVpaalaaabaGaamizaiaadAhadaWgaaWcbaGaamiEaaqabaaakeaacaWGKbGaamiDaaaacqGH9aqpcaWGNbaaaa@4960@  (équation différentielle du mouvement)

b.  Résolution de l’équation différentielle

Conditions initiales : supposons que la position initiale (à l’instant t=0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiDaiabg2da9iaaicdaaaa@3AFE@  ) du solide soit x 0 =0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdaaaa@3BF2@  et sa vitesse initiale soit v x0 =0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4bGaaGimaaqabaGccqGH9aqpcaaIWaaaaa@3CED@ .

Expression de la vitesse :

d v x dt =g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaamaalaaabaGaamizaiaadAhadaWgaaWcbaGaamiEaaqabaaakeaacaWGKbGaamiDaaaacqGH9aqpcaWGNbaaaa@3CCC@

=>

v x =gt+k MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4baabeaakiabg2da9iaadEgacaWG0bGaey4kaSIaam4Aaaaa@3F30@

à t=0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiDaiabg2da9iaaicdaaaa@3AFE@ , v x = v x0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4baabeaakiabg2da9iaadAhadaWgaaWcbaGaamiEaiaaicdaaeqaaaaa@3E57@

=>

g×0+k= v x0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaam4zaiabgEna0kaaicdacqGHRaWkcaWGRbGaeyypa0JaamODamaaBaaaleaacaWG4bGaaGimaaqabaaaaa@41B8@

 

 

=>

k= v x0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaam4Aaiabg2da9iaadAhadaWgaaWcbaGaamiEaiaaicdaaeqaaaaa@3D19@

 

 

et

v x =gt+ v x0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadAhadaWgaaWcbaGaamiEaaqabaGccqGH9aqpcaWGNbGaamiDaiabgUcaRiaadAhadaWgaaWcbaGaamiEaiaaicdaaeqaaaaa@3EAA@

Remarque : si la vitesse initiale est nulle ( v x0 =0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4bGaaGimaaqabaGccqGH9aqpcaaIWaaaaa@3CED@  ), alors l’expression de la vitesse devient v x =gt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadAhadaWgaaWcbaGaamiEaaqabaGccqGH9aqpcaWGNbGaamiDaaaa@3AEA@ .

Expression de la position :

v x = lim Δt0 Δx Δt MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadAhadaWgaaWcbaGaamiEaaqabaGccqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeuiLdqKaamiDaiabgkziUkaaicdaaeqaaOWaaSaaaeaacqqHuoarcaWG4baabaGaeuiLdqKaamiDaaaaaaa@45EF@

=>

v x = dx dt MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadAhadaWgaaWcbaGaamiEaaqabaGccqGH9aqpdaWcaaqaaiaadsgacaWG4baabaGaamizaiaadshaaaaaaa@3CDC@

on en déduit :

dx dt =gt+ v 0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaamaalaaabaGaamizaiaadIhaaeaacaWGKbGaamiDaaaacqGH9aqpcaWGNbGaamiDaiabgUcaRiaadAhadaWgaaWcbaGaaGimaaqabaaaaa@3F57@

 

=>

x= 1 2 g t 2 + v 0 t+k' MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadIhacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadEgacaWG0bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamODamaaBaaaleaacaaIWaaabeaakiaadshacqGHRaWkcaWGRbGaai4jaaaa@4276@

à t=0, x= x 0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiEaiabg2da9iaadIhadaWgaaWcbaGaaGimaaqabaaaaa@3C2B@

=>

x 0 = 1 2 g× 0 2 + v 0 ×0+k' MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadIhadaWgaaWcbaGaaGimaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadEgacqGHxdaTcaaIWaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamODamaaBaaaleaacaaIWaaabeaakiabgEna0kaaicdacqGHRaWkcaWGRbGaai4jaaaa@4716@

 

 

=>

k'= x 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaam4AaiaacEcacqGH9aqpcaWG4bWaaSbaaSqaaiaaicdaaeqaaaaa@3CC8@

 

et

 

x= 1 2 g t 2 + v 0 t+ x 0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadIhacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadEgacaWG0bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamODamaaBaaaleaacaaIWaaabeaakiaadshacqGHRaWkcaWG4bWaaSbaaSqaaiaaicdaaeqaaaaa@42BE@

Remarque : si le solide est lâché du point O ( x 0 =0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdaaaa@3BF2@  ) si la vitesse initiale est nulle ( v x0 =0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FaYxGuc9pgc9q8qqaq=dir=f0=yqaiFf0xe9Fve9Fve9qapdbaqaaeaacaGaaiaabaqaamaaeaqbaaGcbaGaamODamaaBaaaleaacaWG4bGaaGimaaqabaGccqGH9aqpcaaIWaaaaa@3CED@  ), alors l’expression de la position du mobile devient x= 1 2 g t 2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FuK8c8fsY=rqaqpepae9pg0FirpepeYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaadIhacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadEgacaWG0bWaaWbaaSqabeaacaaIYaaaaaaa@3C29@ .