Mouvements de projectiles

 

I.   Eléménts de cinématique

1.  Vecteur vitesse

Soit un mobile animé d’un mouvement quelconque. Soit M la position du mobile à l’instant t et soit M’ la position du mobile à l’instant t’.

Définition : On appelle vecteur vitesse du mobile le vecteur v = lim Δt0 MM' Δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabfs5aejaadshacqGHsgIRcaaIWaaabeaakmaalaaabaWaa8HaaeaacaWGnbGaamytaiaacEcaaiaawEniaaqaaiabfs5aejaadshaaaaaaa@4244@ .

Soit O l’origine du repère. Le vecteur MM' MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGnbGaamytaiaacEcaaiaawEniaaaa@35A9@  peut s’écrire MM' = OM' OM MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGnbGaamytaiaacEcaaiaawEniaiabg2da9maaFiaabaGaam4taiaad2eacaGGNaaacaGLxdcacqGHsisldaWhcaqaaiaad+eacaWGnbaacaGLxdcaaaa@3EFB@ .

Définition : On appelle vecteur position du mobile le vecteur OM MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGpbGaamytaaGaay51Gaaaaa@3500@ .

Remarque : Soit Δ OM MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeuiLdq0aa8HaaeaacaWGpbGaamytaaGaay51Gaaaaa@3666@  le vecteur variation du vecteur position.

Le vecteur vitesse du mobile peut s’écrire v = lim Δt0 MM' Δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabfs5aejaadshacqGHsgIRcaaIWaaabeaakmaalaaabaWaa8HaaeaacaWGnbGaamytaiaacEcaaiaawEniaaqaaiabfs5aejaadshaaaaaaa@4244@

 => v = lim Δt0 OM' OM Δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabfs5aejaadshacqGHsgIRcaaIWaaabeaakmaalaaabaWaa8HaaeaacaWGpbGaamytaiaacEcaaiaawEniaiabgkHiTmaaFiaabaGaam4taiaad2eaaiaawEniaaqaaiabfs5aejaadshaaaaaaa@468D@
 => v = lim Δt0 Δ OM Δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabfs5aejaadshacqGHsgIRcaaIWaaabeaakmaalaaabaGaeuiLdq0aa8HaaeaacaWGpbGaamytaaGaay51GaaabaGaeuiLdqKaamiDaaaaaaa@4301@
 => v = d OM dt MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaSaaaeaacaWGKbWaa8HaaeaacaWGpbGaamytaaGaay51GaaabaGaamizaiaadshaaaaaaa@39ED@

Coordonnées du vecteur v MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaaaaa@32B3@  :
Soit OM MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGpbGaamytaaGaay51Gaaaaa@3500@  de coordonnées x,y,z MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamiEaiaacYcacaaMc8UaaGPaVlaadMhacaGGSaGaaGPaVlaaykW7caWG6baaaa@3C2C@  dans le repère (O, i , j , k ) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaaiikaiaad+eacaGGSaGabmyAayaalaGaaiilaiqadQgagaWcaiaacYcaceWGRbGbaSaacaGGPaaaaa@38E6@ . Le vecteur position a pour expression : OM =x i +y j +z k MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGpbGaamytaaGaay51GaGaeyypa0JaamiEaiqadMgagaWcaiabgUcaRiaadMhaceWGQbGbaSaacqGHRaWkcaWG6bGabm4Aayaalaaaaa@3DC7@ .

Le vecteur vitesse du mobile peut s’écrire v = d(x i +y j +z k ) dt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaSaaaeaacaWGKbGaaiikaiaadIhaceWGPbGbaSaacqGHRaWkcaWG5bGabmOAayaalaGaey4kaSIaamOEaiqadUgagaWcaiaacMcaaeaacaWGKbGaamiDaaaaaaa@3FAE@

 =>

v = dx dt i + dy dt j + dz dt k MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaSaaaeaacaWGKbGaamiEaaqaaiaadsgacaWG0baaaiqadMgagaWcaiabgUcaRmaalaaabaGaamizaiaadMhaaeaacaWGKbGaamiDaaaaceWGQbGbaSaacqGHRaWkdaWcaaqaaiaadsgacaWG6baabaGaamizaiaadshaaaGabm4Aayaalaaaaa@440B@
v = x . i + y . j + z . k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaCbiaeaacaWG4baaleqabaGaeSigI8gaaOGaaiOlaiqadMgagaWcaiabgUcaRmaaxacabaGaamyEaaWcbeqaaiablIHiVbaakiaac6caceWGQbGbaSaacqGHRaWkdaWfGaqaaiaadQhaaSqabeaacqWIyiYBaaGccaGGUaGabm4Aayaalaaaaa@4236@

 

 

On remarquera que le vecteur vitesse est tangent à la trajectoire.

2.  Vecteur accélération :

Nous savons déjà que le vecteur accélération a pour expression a = d v dt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmyyayaalaGaeyypa0ZaaSaaaeaacaWGKbWaa8HaaeaacaWG2baacaGLxdcaaeaacaWGKbGaamiDaaaaaaa@392E@  (voir mécanique de Newton).

Le vecteur accélération du mobile peut s’écrire a = d( v x i + v y j + v z k ) dt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmyyayaalaGaeyypa0ZaaSaaaeaacaWGKbGaaiikaiaadAhadaWgaaWcbaGaamiEaaqabaGcceWGPbGbaSaacqGHRaWkcaWG2bWaaSbaaSqaaiaadMhaaeqaaOGabmOAayaalaGaey4kaSIaamODamaaBaaaleaacaWG6baabeaakiqadUgagaWcaiaacMcaaeaacaWGKbGaamiDaaaaaaa@432C@

 =>

a = d v x dt i + d v y dt j + d v z dt k MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmyyayaalaGaeyypa0ZaaSaaaeaacaWGKbGaamODamaaBaaaleaacaWG4baabeaaaOqaaiaadsgacaWG0baaaiqadMgagaWcaiabgUcaRmaalaaabaGaamizaiaadAhadaWgaaWcbaGaamyEaaqabaaakeaacaWGKbGaamiDaaaaceWGQbGbaSaacqGHRaWkdaWcaaqaaiaadsgacaWG2bWaaSbaaSqaaiaadQhaaeqaaaGcbaGaamizaiaadshaaaGabm4Aayaalaaaaa@4789@
a = d 2 x d t 2 i + d 2 y d t 2 j + d 2 z d t 2 k MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmyyayaalaGaeyypa0ZaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaamiEaaqaaiaadsgacaWG0bWaaWbaaSqabeaacaaIYaaaaaaakiqadMgagaWcaiabgUcaRmaalaaabaGaamizamaaCaaaleqabaGaaGOmaaaakiaadMhaaeaacaWGKbGaamiDamaaCaaaleqabaGaaGOmaaaaaaGcceWGQbGbaSaacqGHRaWkdaWcaaqaaiaadsgadaahaaWcbeqaaiaaikdaaaGccaWG6baabaGaamizaiaadshadaahaaWcbeqaaiaaikdaaaaaaOGabm4Aayaalaaaaa@49A8@
a = x . i + y . j + z . k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGHbaacaGLxdcacqGH9aqpdaWfGaqaaiaadIhaaSqabeaacqWIyiYBcqWIyiYBaaGccaGGUaGabmyAayaalaGaey4kaSYaaCbiaeaacaWG5baaleqabaGaeSigI8MaeSigI8gaaOGaaiOlaiqadQgagaWcaiabgUcaRmaaxacabaGaamOEaaWcbeqaaiablIHiVjablIHiVbaakiaac6caceWGRbGbaSaaaaa@4771@

 

 

 

 

II.  Mouvement d’un projectile dans le champ de pesanteur uniforme

Soit un objet S lancé avec une vitesse initiale v o MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWG2bWaaSbaaSqaaiaad+gaaeqaaaGccaGLxdcaaaa@357F@  dans le champ de pesanteur supposé localement uniforme.

1.  Vecteur accélération :

Système étudié : l’objet S.

Référentiel : terrestre considéré comme galiléen (la durée du mouvement est faible par rapport à la durée d’un jour).

Forces extérieures exercées sur l’objet S : la force de pesanteur (ou poids) P MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmiuayaalaaaaa@328D@ .

Application de la deuxième loi de Newton : F ext =m. a G MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaabqaeaadaWhcaqaaiaadAeadaWgaaWcbaGaamyzaiaadIhacaWG0baabeaaaOGaay51GaaaleqabeqdcqGHris5aOGaeyypa0JaamyBaiaac6cadaWhcaqaaiaadggadaWgaaWcbaGaam4raaqabaaakiaawEniaaaa@3F9F@

 

P =m. a G MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeyO0H4TabmiuayaalaGaeyypa0JaamyBaiaac6cadaWhcaqaaiaadggadaWgaaWcbaGaam4raaqabaaakiaawEniaaaa@3B2F@
m. g =m. a G MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeyO0H4TaamyBaiaac6cadaWhcaqaaiaadEgaaiaawEniaiabg2da9iaad2gacaGGUaWaa8HaaeaacaWGHbWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcaaaa@3E8C@
a G = g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeyO0H49aa8HaaeaacaWGHbWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcacqGH9aqpdaWhcaqaaiaadEgaaiaawEniaaaa@3B44@

2.  Equations horaires paramétriques :

Conditions initiales : Supposons qu’à l’instant t=0, le mobile est lancé de l’origine du repère O avec une vitesse initiale v 0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWG2bWaaSbaaSqaaiaaicdaaeqaaaGccaGLxdcaaaa@3545@  faisant un angle α MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaeqySdegaaa@3345@  avec l’axe Oy.

Le vecteur position initiale s’écrit alors O G 0 = 0 O G 0 { x 0 =0 y 0 =0 z 0 =0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGpbGaam4ramaaBaaaleaacaaIWaaabeaaaOGaay51GaGaeyypa0Zaa8HaaeaacaaIWaaacaGLxdcacqGHuhY2daWhcaqaaiaad+eacaWGhbWaaSbaaSqaaiaaicdaaeqaaaGccaGLxdcadaGabaabaeqabaGaamiEamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdaaeaacaWG5bWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaaGimaaqaaiaadQhadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIWaaaaiaawUhaaaaa@4C29@ .

Le vecteur vitesse initiale a pour coordonnées v 0 { v 0x =0 v 0y = v 0 .cosα v 0z = v 0 .sinα MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWG2bWaaSbaaSqaaiaaicdaaeqaaaGccaGLxdcadaGabaabaeqabaGaamODamaaBaaaleaacaaIWaGaamiEaaqabaGccqGH9aqpcaaIWaaabaGaamODamaaBaaaleaacaaIWaGaamyEaaqabaGccqGH9aqpcaWG2bWaaSbaaSqaaiaaicdaaeqaaOGaaiOlaiGacogacaGGVbGaai4Caiabeg7aHbqaaiaadAhadaWgaaWcbaGaaGimaiaadQhaaeqaaOGaeyypa0JaamODamaaBaaaleaacaaIWaaabeaakiaac6caciGGZbGaaiyAaiaac6gacqaHXoqyaaGaay5Eaaaaaa@5110@ .

Coordonnées du vecteur vitesse :

D’après le paragraphe 1., les coordonnées du vecteur accélération sont a G { a x =0 a y =0 a z =g MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGHbWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcadaGabaabaeqabaGaamyyamaaBaaaleaacaWG4baabeaakiabg2da9iaaicdaaeaacaWGHbWaaSbaaSqaaiaadMhaaeqaaOGaeyypa0JaaGimaaqaaiaadggadaWgaaWcbaGaamOEaaqabaGccqGH9aqpcqGHsislcaWGNbaaaiaawUhaaaaa@4311@ .

Or a G = d v dt MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGHbWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcacqGH9aqpdaWcaaqaaiaadsgaceWG2bGbaSaaaeaacaWGKbGaamiDaaaaaaa@3A2F@  ; par intégration, on obtient : v 0 { v x = v 0x v y = v 0y v z =gt+ v 0z v 0 { v x =0 v y = v 0 cosα v z =gt+ v 0 sinα MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6E5D@

Le mouvement est uniforme selon l’axe Oy et uniformément varié selon l’axe Oz.

Coordonnées du vecteur position :

D’après le paragraphe I, le vecteur vitesse est la dérivée du vecteur position : v G = d OG dt MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWG2bWaaSbaaSqaaiaadEeaaeqaaaGccaGLxdcacqGH9aqpdaWcaaqaaiaadsgadaWhcaqaaiaad+eacaWGhbaacaGLxdcaaeaacaWGKbGaamiDaaaaaaa@3C8B@  ;

Par intégration, on obtient : OG { x= x 0 y= v 0 cosα.t+ y 0 z= 1 2 g t 2 + v 0 sinα.t+ z 0 OG { x=0 y= v 0 cosα.t z= 1 2 g t 2 + v 0 sinα.t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7DD3@

Le mouvement s’effectue dans le plan (O,y,z).

3.  Equation de la trajectoire :

t= y v 0 cosα MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamiDaiabg2da9maalaaabaGaamyEaaqaaiaadAhadaWgaaWcbaGaaGimaaqabaGcciGGJbGaai4BaiaacohacqaHXoqyaaaaaa@3B10@  et z= 1 2 g ( y v 0 cosα ) 2 + v 0 sinα. y v 0 cosα MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamOEaiabg2da9maalaaabaGaeyOeI0IaaGymaaqaaiaaikdaaaGaam4zaiaacIcadaWcaaqaaiaadMhaaeaacaWG2bWaaSbaaSqaaiaaicdaaeqaaOGaci4yaiaac+gacaGGZbGaeqySdegaaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG2bWaaSbaaSqaaiaaicdaaeqaaOGaci4CaiaacMgacaGGUbGaeqySdeMaaiOlamaalaaabaGaamyEaaqaaiaadAhadaWgaaWcbaGaaGimaaqabaGcciGGJbGaai4BaiaacohacqaHXoqyaaaaaa@5023@ .

On en déduit l’équation de la trajectoire du centre d’inertie du mobile :

z= g 2 v 0 2 cos 2 α y 2 +tanα.y MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamOEaiabg2da9maalaaabaGaeyOeI0Iaam4zaaqaaiaaikdacaWG2bWaa0baaSqaaiaaicdaaeaacaaIYaaaaOGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySdegaaiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkciGG0bGaaiyyaiaac6gacqaHXoqycaGGUaGaamyEaaaa@474F@

La trajectoire est un arc de parabole.

Remarques :

On appelle « portée du tir » la distance OP et « flèche » la hauteur h.

Au point P, z=0 g 2 v 0 2 cos 2 α y 2 +tanα.y=0y( g 2 v 0 2 cos 2 α y+tanα)=0| y=0 ou y= 2 v 0 2 sinα.cosα g = v 0 2 sin2α g MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@86BB@

Au point S, le vecteur vitesse v S MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWG2bWaaSbaaSqaaiaadofaaeqaaaGccaGLxdcaaaa@3563@  est horizontal et sa coordonnée selon Oz est nulle ( v Sz =0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGtbGaamOEaaqabaGccqGH9aqpcaaIWaaaaa@366E@  ).