96
2 Particle Dynamics
Setting α = −
dm
dt
and
dv
dt
= 0, minimum exhaust velocity
v r =
g
α
(b) Dividing (1) by m
dv
dt
= −
v r
m
dm
dt
− g
(2)
∴ dv = −v r
dm
m
− g dt
Assuming that v r and g remain constant, and at t = 0, v = 0 and m = m 0 ,
v B
0
dv = −v r
m B
m 0
dm
m
− g
t
0
dt
∴ v B = −v r ln
m 0
m B
− gt
(3)
where m 0 is the initial mass of the system and m B the mass at burn-out
velocity v B
(c) Setting g = 0 in (2)
dv
dt
= −
v r
m
dm
dt
(4)
−
dm
m
= α = Positive constant
dm = −αdt
∴
dm = −α
dt + C
where C is the constant of integration
m = −αt + C
When t = 0, m = m 0 . Therefore, C = m 0
m(t) = m 0 − αt
(5)