Rocket Flight. A model rocket having initial mass
m
0
k
g
is launched vertically from the ground. The rocket expels gas at a constant rate of
α
k
g
/
s
e
c
and at a constant velocity of
β
m
/
s
e
c
relative to the rocket. Assume that the magnitude of the gravitational force is proportional to the mass with proportionality constant
g
. Because the mass is not constant, Newton’s second law leads to the equation
(
m
0
−
α
t
)
d
v
d
t
−
α
β
≡
−
g
(
m
0
−
α
t
)
,
where
v
≡
d
x
/
d
t
is the velocity of the rocket,
x
is its height above the ground, and
m
0
−
α
t
is the mass of the rocket at
t
sec
after launch. If the initial velocity is zero, solve the above equation to determine the velocity of the rocket and its height above ground for
0
≤
t
<
m
0
/
α
.