A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let
v
=
v
(
t
)
be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let
M
=
M
(
t
)
be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that
F
=
M
d
v
d
t
−
u
b
where the force
F
=
−
M
g
. Thus
1
M
d
v
d
t
−
u
b
=
−
M
g
Let
M
i
be the mass of the rocket without fuel,
M
2
the initial mass of the fuel, and
M
0
=
M
1
+
M
2
. Then, until the fuel runs out at time
t
=
M
2
/
b
, the mass is
M
=
M
0
−
b
t
.
(a) Substitute
M
=
M
0
−
b
t
into Equation 1 and solve the resulting equation for v. Use the initial condition
v
(
0
)
=
0
to evaluate the constant.
(b) Determine the velocity of the rocket at time
t
=
M
2
/
b
. This is called the burnout velocity.
(c) Determine the height of the rocket
y
=
y
(
t
)
at the burnout time.
(d) Find the height of the rocket at any time t.