Introduction 1 Présentation de la fusée

L
S s m0
ROz R0
O0z0
z h(t)
R0u=−uez
P(t)P
h0=h(t= 0) P0=P(t= 0)
s S L m0h0P0P ρ
3 cm275 cm227 cm 140 g 2 cm 3 bar 1 bar 1000 kg.m−3

10%
h(0) = h0<< L
P(t)
h0= 7 cm
S(L−h0)P0
S(L−h)P
P(t)[S¡L−h(t))]γ=P0[S(L−h0)]γ
P(t) = P0³1−γh0−h(t)
L´
−grad P
ρg
η∆v
−ρae
ρ∂v
∂t +ρ(v.grad v)v=−grad P+ρg−ρae−η∆v
h0/cScS≈1000 m.s−1
h0/cS≈2.10−5s
|grad P| ∼ P0−P
h0
= 1.107N.m−3

ρg ≈104N.m−3
|η∆v| ≈ ηV/(√s)2 V
√s
|η∆v| ∼ 1.10−330
(1.10−2)2= 300 N.m−3
h
1
2ρ˙
h2+P(t) = 1
2ρu2+P
Dm=su =−S˙ h
S >> s ˙ h << u
u=−S
s˙
h=s2¡P(t)−P¢
ρ
h
−S
s˙
h=s2
ρ³P0³1−γh0−h(t) L´−P´
˙
h=−s
Ss2(P0−P) ρ³1−γP0 P0−P
h0−h(t)
L´
(h << L)
˙
h=−s
Ss2(P0−P) ρ³1−γ 2
P0
P0−P
h0−h(t)
L´
˙
h+s
S
γP0
2Ls2 ρ(P0−P)h=−s Ss2(P0−P) ρ³1−γ 2
P0
P0−P
h0
L´
h/L s/S
h0/L
˙
h+s
S
γP0
2Ls2 ρ(P0−P)h=−s Ss2(P0−P) ρ)

h(t)
1
τ=s S
γP0
2Lq2 ρ(P0−P)hF= 2LP0−P γP0
h(0) = h0
h(t) = −hF+ (h0+hF)e−t/τ
h(0) = 0
τ=τln hF+h0
hF
τ= 0,32 s hF= 26 cm h0<< hF
τ=τh0
hF=S sh0qρ 2(P0−P)= 0,025 s
τ h(t)
h(t) = h0¡1−t
τ¢
ROz
m(t)¨z=−m(t)g+Dmu
m(t) = m0+ρSh(t)t Dm=−˙m
u
Dm=−ρS ˙ h=ρsu
h(t)τ˙ h≈C
u≈s2(P0−P)
ρ
m(t)¨z=−m(t)g+ρsu2≈ −m(t)g+ 2(P0−P)s
m(t)g≤10 N 2(P0−P)s≈120 N
m(t)g2(P0−P)s
m(t)¨z= 2(P0−P)s

˙z(t)
m0
m(t) = m0+ρSh0³1−t
τ´
¨z=2(P0−P)s
m0+ρSh0(1 −t
τ)
t= 0 t=τ
˙z=q2(P0−P)
ρln m0+ρSh0 m0= 15 m.s−1
3 bars
P > P
h0
L= 1 −³P P0´1/γ ≈0,5
h0
L≈0,3
S/s
ρae≈ρ˙z
τ= 6.105N.m−3|grad P| ≈ P0−P h0
= 1.107N.m−3