KAHIBARO
Table of Contents
Changing Mass in Mechanics
In many mechanics problems, the mass of an object stays constant. A thrown ball, a sliding block, or a falling stone usually keeps the same mass while we study its motion. A variable-mass system is different. Its mass changes with time because matter enters or leaves the system.
A rocket is the most famous example. As it burns fuel, hot gases are expelled backward, so the rocket loses mass. Other examples include a leaking cart, a conveyor belt receiving sand, and a raindrop growing as water condenses onto it. In all these cases, the usual form of Newton's second law, $F = ma$, must be used carefully, because the mass is not constant.
Why Ordinary Forms Need Care
For constant mass, momentum is
$$
p = mv
$$
and Newton's second law is often written as
$$
F_{\text{net}} = ma
$$
or equivalently,
$$
F_{\text{net}} = \frac{dp}{dt}
$$
When mass changes, the momentum of the system is still $p = mv$, but now both $m$ and $v$ may change. Then
$$
\frac{dp}{dt} = \frac{d}{dt}(mv) = m\frac{dv}{dt} + v\frac{dm}{dt}
$$
This expression is mathematically true, but it must be interpreted carefully in physics. The reason is that when mass enters or leaves the system, that mass carries momentum with it. So we cannot simply take $F = \frac{d}{dt}(mv)$ for the chosen object alone without accounting for the momentum flow across the system boundary.
For a variable-mass system, you must keep track of mass entering or leaving the system and the momentum that this mass carries. Using $F = ma$ without this care can give wrong results.
Choosing the System
The key idea is to define what belongs to the system at each moment. Suppose we choose the rocket alone as the system. A short time later, some fuel that was part of the rocket has been expelled and is no longer in the system. Since mass has crossed the system boundary, we must include the effect of that moving mass.
This is the main new feature of variable-mass systems. The change in momentum of the chosen object is affected not only by external forces, but also by the momentum carried by the mass transfer.
Momentum Balance for a Small Time Interval
Consider an object moving with velocity $v$ at some instant and having mass $m$. During a short time interval $dt$, it ejects a small amount of mass $dm$. For a rocket, the rocket loses mass, so $dm < 0$ if $m$ refers to the rocket's mass change.
To avoid sign confusion, it is often easier to speak of a positive expelled mass $d m_{\text{out}} > 0$, while the rocket mass changes by
$$
dm = -d m_{\text{out}}
$$
Let the expelled mass leave with speed $u$ relative to the rocket. If the rocket moves forward, the exhaust is usually backward relative to the rocket.
We compare total momentum before and after the small interval. External forces may also act, such as gravity. The change in total momentum equals the impulse of external forces.
This careful momentum accounting leads to the rocket equation, which is treated in the next chapter. Here the important point is the method, the system changes mass, so momentum flow must be included.
General Idea of Momentum Flux
A useful way to think about variable-mass systems is that mass crossing the boundary transports momentum.
If mass enters the system with velocity $\vec{v}_{\text{in}}$ and rate $\frac{dm_{\text{in}}}{dt}$, it brings momentum in. If mass leaves with velocity $\vec{v}_{\text{out}}$ and rate $\frac{dm_{\text{out}}}{dt}$, it takes momentum out.
In one dimension, the momentum balance can be written schematically as
$$
F_{\text{ext}} = \frac{d}{dt}(mv) - v_{\text{in}}\frac{dm_{\text{in}}}{dt} + v_{\text{out}}\frac{dm_{\text{out}}}{dt}
$$
The exact sign convention depends on how the system and directions are defined, but the physical meaning is always the same, momentum carried by incoming and outgoing mass must be included.
A changing mass alone does not create motion. Motion changes because mass transfer carries momentum, and because external forces may act.
Mass Loss and Mass Gain
Some systems lose mass, others gain mass. The physical behavior depends strongly on how that transferred mass moves.
Consider the following comparison.
| Situation | Mass change | Important effect |
|---|---|---|
| Rocket expelling fuel | Mass decreases | Outgoing mass carries backward momentum |
| Leaking bucket | Mass decreases | Lost mass may leave with similar velocity to bucket |
| Cart catching rain | Mass increases | Incoming mass may initially have different velocity |
| Conveyor receiving sand | Mass increases | New mass can slow the system if it arrives with lower speed |
A system gaining mass can slow down even without a horizontal external force, simply because the incoming mass may have little or no horizontal momentum and must be accelerated by the system.
A system losing mass does not automatically speed up. It speeds up only if the expelled mass leaves with relative velocity that produces thrust.
Example Idea, Cart Collecting Sand
Imagine a cart moving horizontally with speed $v$ while sand falls vertically into it. Suppose there is no horizontal external force.
Before entering the cart, the sand has no horizontal momentum. After entering, it moves with the cart, so the cart must give the sand horizontal momentum. As a result, the cart's speed decreases.
If the mass of the cart plus collected sand is $m$, then conservation of horizontal momentum gives the basic idea
$$
mv = \text{constant}
$$
provided no external horizontal force acts.
This means that as $m$ increases, $v$ decreases.
This is a classic example of a variable-mass system where the main effect is not propulsion, but slowing due to added mass.
Example Idea, Leaking System
Now imagine a cart with sand leaking through a hole in the bottom. If the sand leaves with the same horizontal velocity as the cart at the moment it leaves, then the leak itself does not provide horizontal thrust. In that case, if there is no horizontal external force, the cart's horizontal speed may remain unchanged.
This may seem surprising at first. Losing mass does not by itself push an object. A push appears only if the departing mass has a different velocity from the main body in the direction of motion.
Relative Velocity Matters
The central quantity in rocket-type motion is the velocity of transferred mass relative to the main body. If exhaust leaves a rocket backward relative to the rocket, then the rocket gains forward momentum.
If the exhaust velocity relative to the rocket is $u$, then the exhaust velocity in a ground frame is not simply $u$. It must be combined with the rocket's own velocity. This is why relative motion is essential in variable-mass problems.
For a rocket moving in one dimension, if forward is positive and exhaust is ejected backward with speed $u$ relative to the rocket, then the exhaust velocity in the ground frame is typically
$$
v_{\text{exhaust}} = v - u
$$
where $u > 0$ is the exhaust speed relative to the rocket.
This relative-velocity idea is the source of rocket thrust.
External Forces in Variable-Mass Motion
Variable-mass systems are often also subject to external forces. For rockets near Earth, gravity acts downward. For vehicles moving through air, drag may also matter. For carts and sand, friction may act.
So momentum balance usually has two contributions. One is the effect of external forces. The other is the momentum carried by transferred mass.
In a short time $dt$, the impulse of external forces is
$$
F_{\text{ext}}\,dt
$$
and this must equal the change in total momentum of all relevant parts.
This makes variable-mass motion an extension of momentum methods, not a replacement of them.
Visualizing a Rocket as an Open System
A rocket can be viewed as an open system, meaning matter crosses its boundary.
The dashed rectangle shows a chosen boundary around the rocket. Exhaust crosses this boundary and carries momentum away. That is why the rocket's changing mass must be treated with special care.
Common Sign Conventions
Sign conventions are a major source of mistakes. It helps to set them clearly at the beginning.
| Quantity | Common convention |
|---|---|
| Forward direction | Positive |
| Rocket velocity $v$ | Positive if rocket moves forward |
| Rocket mass change $dm$ | Negative when fuel is expelled |
| Exhaust speed relative to rocket $u$ | Often taken as a positive magnitude |
| Exhaust velocity in ground frame | Often $v - u$ |
Because different books define $u$ differently, always check whether $u$ means a signed velocity or just a positive speed magnitude.
In rocket problems, define the sign of $v$, $u$, and $dm$ before writing equations. Many wrong answers come from sign errors, not physics errors.
What Makes Variable-Mass Systems Special
The unique feature of these systems is that momentum can change through two routes. External forces can act, and mass crossing the boundary can carry momentum in or out.
That is why variable-mass systems are often called open systems in mechanics. The object we track is not closed, because matter exchange occurs.
For beginners, the safest strategy is this. Write down the system clearly, identify whether mass enters or leaves, determine the velocity of that mass in the chosen reference frame, and then apply momentum conservation or momentum balance over a short time interval.
Link to Rocket Motion
Variable-mass systems provide the foundation for rocket motion. Once we understand that ejected mass carries momentum and creates thrust, we can derive the rocket equation and study how a rocket speeds up as it burns fuel.
That full derivation belongs to the next chapter. Here the essential lesson is simpler. When mass changes, mechanics must account for momentum transfer through the system boundary.
Core idea of variable-mass systems: use momentum balance, not just $F = ma$ in its simplest constant-mass form.
Summary
A variable-mass system is one whose mass changes because matter enters or leaves. In such systems, the changing motion cannot be described correctly unless we include the momentum carried by the transferred mass. A system can lose mass, gain mass, speed up, slow down, or keep the same speed, depending on the velocity of the exchanged mass and on any external forces. This idea is the basis for understanding rockets, fuel burn, and many other real physical systems.