There are certain system in which mass is redistributed with the surrounding. The change in mass of the system results as there is exchange of mass between the system and its surrounding. The motion of a leaking balloon, as shown, is an example of changing mass system.
Similar is the situation in the case of a rocket. Now, does this exchange of mass with surrounding have any bearing on the form of Newton’s second law or about the meaning of force as product of mass and acceleration? This is the question that we need to answer. To investigate this question, let us consider the motion of the rocket. It acquires very high speed quickly as a result of high speed gas escaping in the opposite direction.
Applying Newton’s second law, considering that the mass of the system is changing :
F
=
ⅆ
(
m
v
)
ⅆ
t
⇒
F
=
m
ⅆ
v
ⅆ
t
+
v
ⅆ
m
ⅆ
t
⇒
F
=
m
a
+
v
ⅆ
m
ⅆ
t
F
=
ⅆ
(
m
v
)
ⅆ
t
⇒
F
=
m
ⅆ
v
ⅆ
t
+
v
ⅆ
m
ⅆ
t
⇒
F
=
m
a
+
v
ⅆ
m
ⅆ
t
This apparent mass variant form of Newton’s second law appears to destroy the well founded meaning of force as "product of invariant mass and acceleration". As a matter of fact, it is not so.
Let us recall that the relation, F = ma, essentially underlines relation between force (cause) and acceleration (effect). Now look closely at the additional term resulting from change in mass in the context of a real time case like of a rocket. Rearranging, we have :
⇒
F

v
ⅆ
m
ⅆ
t
=
m
a
⇒
F

v
ⅆ
m
ⅆ
t
=
m
a
Here, we need to be cautious in interpreting this expression. The first question that we need to answer is : what does "v" represent in the expression? Is it the velocity of rocket or escaping gas or is it the relative velocity of rocket with respect to escaping gas? The second question, then, is what does "a" represent?
We see that "v" should represent the velocity of the rocket in the ground inertial reference. This is the way we interpret Newton's second law for a constant mass body? The situation here, however, is different in that a part of the body is continuously being transferred to other system, which itself is moving. In order to simplify the analysis, we consider that rocket is ejecting gas at constant rate and with constant relative velocity. We can then say that ejected gas is applying force on the rocket in the reference of ejected gas, which is moving at uniform velocity. As such moving mass of gas constitute an inertial frame in which force is applied. The velocity, "v", therefore, represents the velocity of the rocket with respect to ejected mass of gas. We, then, rewrite the equation as :
⇒
F

v
r
ⅆ
m
ⅆ
t
=
m
a
⇒
F

v
r
ⅆ
m
ⅆ
t
=
m
a
where "
v
r
v
r
" represents the velocity of rocket with respect to gas being ejected.
Now, we turn to answer second question about acceleration. We know that acceleration is time rate of change of velocity. But, as we discussed, we measure velocity of the rocket with reference to ejected gas. Therefore, it follows that rate of change should also be associated with relative velocity  not the absolute velocity with respect to ground. However, there is an important difference between velocity and acceleration, as measured in two inertial frames viz. ground and ejected gas. Though, measurement of velocities are different, but measurement of "change" in velocity remains same in all inertial frames. Hence, we can interpret "a" as acceleration measured in either of two references without any distinction.
Now, we are in position to correctly interpret the additional term "

v
r
ⅆ
m
ⅆ
t

v
r
ⅆ
m
ⅆ
t
" by answering the following question : "How could a high speed ejection of gas make the rocket accelerate in the absence of any other external force? We should remember that gravitational pull and air resistance, as a matter of fact, retards the motion of rocket. The answer lies in the fact that the additional term "

v
r
ⅆ
m
ⅆ
t

v
r
ⅆ
m
ⅆ
t
" actually represents a force called "thrust" on the rocket. For rocket to accelerate, this thrust is an external force on the rocket. If we represent thrust by symbol "T", then :
⇒
T
=

v
r
ⅆ
m
ⅆ
t
⇒
T
=

v
r
ⅆ
m
ⅆ
t
Substituting in the equation of Newton's law,
⇒
F
+
T
=
m
a
⇒
F
+
T
=
m
a
The additional term, therefore, is not an “effect” term like “ma”, but a "cause" term representing “external force”, which results from the exchange of mass with the surrounding.
Clearly, there is no other external force other than thrust that gives such a great acceleration to a rocket. Gravitational and air resistance actually retards the motion of a rocket moving vertically upward. Thus, we can safely say that thrust is the only force that imparts acceleration in the direction of the motion of the rocket. In case rocket is fired from a region where other external forces like that of gravity and air resistance can be neglected, then we can put , F = 0. Then,
⇒
T
=
m
a
⇒
T
=
m
a
In the nutshell, we see that a varying mass does not change the fundamental aspect of Newton’s second law of motion. It only modifies the external force on the body. The net external force is still equal to the product of mass and acceleration.
We shall reinforce these concepts with another approach that makes use of conservation of linear momentum in a separate module on rocket.
Problem : Air from a leaking balloon of mass 15 gm gushes out at a constant speed of 10 m/s. The balloon shrinks completely in 10 seconds and reduces to a mass of 5 gm. Find the average force on the balloon.
Solution : There is no external force on balloon initially. However, gushing air constitutes an exchange of mass between balloon and its surrounding. This generates an external force on the balloon given by :
F
avg
=

v
r
x
Δ
m
Δ
t
⇒
F
avg
=

10
x
0.005

0.015
10
⇒
F
avg
=
0.01
N
F
avg
=

v
r
x
Δ
m
Δ
t
⇒
F
avg
=

10
x
0.005

0.015
10
⇒
F
avg
=
0.01
N
We summarize the discussion held, in the context of classical mechanics, so far as :
 Given mass is invariant.
 In some instances, the the variance of the mass of body system observed is actually a redistribution of mass from one body system to more than one body systems.
 Irrespective of the nature of mass of a body system (varying or constant), external force is equal to time rate change of momentum.
 If there is redistribution of mass, then it results into an external force on the original body, modifying the external force on it.
 Irrespective of the nature of mass of a body system (varying or constant), the net external force is equal to the product of mass and acceleration.
 Irrespective of the nature of mass of a body system (varying or constant), momentum and acceleration forms of Newton’s second law are equivalent and consistent to each other.