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
Δt→0
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
Δt→0
MM'
→
Δt
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabfs5aejaadshacqGHsgIRcaaIWaaabeaakmaalaaabaWaa8HaaeaacaWGnbGaamytaiaacEcaaiaawEniaaqaaiabfs5aejaadshaaaaaaa@4244@
|
=>
v
→
=
lim
Δt→0
OM'
→
−
OM
→
Δt
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabfs5aejaadshacqGHsgIRcaaIWaaabeaakmaalaaabaWaa8HaaeaacaWGpbGaamytaiaacEcaaiaawEniaiabgkHiTmaaFiaabaGaam4taiaad2eaaiaawEniaaqaaiabfs5aejaadshaaaaaaa@468D@
=>
v
→
=
lim
Δt→0
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=0⇒y(
−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@
).