6Systems of particles

IA Dynamics and Relativity

6.4 Variable-mass problem
All systems we’ve studied so far have fixed mass. However, in real life, many
objects have changing mass, such as rockets, fireworks, falling raindrops and
rolling snowballs.
Again, we will use Newton’s second law, which states that
dp
dt
= F, with p = m
˙
r.
We will consider a rocket moving in one dimension with mass
m
(
t
) and velocity
v
(
t
). The rocket propels itself forwards by burning fuel and ejecting the exhaust
at velocity u relative to the rocket.
At time t, the rocket looks like this:
m(t)
v(t)
At time
t
+
δt
, it ejects exhaust of mass
m
(
t
)
m
(
t
+
δt
) with velocity
v
(
t
)
u + O(δt).
m(t)
v(t)
m
v(t) u
The change in total momentum of the system (rocket + exhaust) is
δp = m(t + δt)v(t + δt) + [m(t) m(t + δt)][v(t) u(t) + O(δt)] m(t)v(t)
= (m + ˙t + O(δt
2
))(v + ˙vδt + O(δt
2
)) ˙t(v u) + O(δt
2
) mv
= ( ˙mv + m ˙v ˙mv + ˙mu)δt + O(δt
2
)
= (m ˙v + ˙mu)δt + O(δt
2
).
Newton’s second law gives
lim
δ0
δp
δt
= F
where F is the external force on the rocket. So we obtain
Proposition (Rocket equation).
m
dv
dt
+ u
dm
dt
= F.
Example.
Suppose that we travel in space with
F
= 0. Assume also that
u
is
constant. Then we have
m
dv
dt
= u
dm
dt
.
So
v = v
0
+ u log
m
0
m(t)
,
Note that we are expressing things in terms of the mass remaining
m
, not time
t
.
Note also that the velocity does not depend on the rate at which mass is
ejected, only the velocity at which it is ejected. Of course, if we expressed
v
as a
function of time, then the velocity at a specific time does depend on the rate at
which mass is ejected.
Example.
Consider a falling raindrop of mass
m
(
t
), gathering mass from a
stationary cloud. In this case, u = v. So
m
dv
dt
+ v
dm
dt
=
d
dt
(mv) = mg,
with
v
measured downwards. To obtain a solution of this, we will need a model
to determine the rate at which the raindrop gathers mass.