Momentum Dynamics

What is Momentum?

The momentum of an object is defined by: $$ \vec{p} = m \vec{v} $$

$ \vec{p} $ is the momentum, expressed in kg·m/s,
$ m $ is the mass of the object in kg,
$ \vec{v} $ is the velocity of the object in m/s.

Motion in Classical Mechanics

In classical mechanics, Newton's second law represents the net force applied to an object: $$ \vec{F} = m \vec{a} $$ This law describes how force acts on an object to change its motion by producing acceleration. It is valid in inertial reference frames (non-accelerated) and for speeds much lower than the speed of light.

However, this formulation assumes that the mass $ m $ of the object is constant. For a more general description of dynamics, especially in systems where mass varies (like a rocket), it is necessary to use a more fundamental approach based on momentum $ \vec{p} $.

Momentum Theorem

"In an inertial reference frame, the time derivative of the momentum of a system is equal to the sum of the external forces applied to that system."

$$ \frac{d\vec{p}}{dt} = \sum \vec{F} $$

$ \vec{p} $ is the momentum of the system.
dt denotes an infinitesimal element of time, i.e., a very small variation in time.
sum(F) represents the external forces.

General Formulation of the Principle of Dynamics

Momentum dynamics is a fundamental concept in physics that describes how forces act on a system to change its motion.

The equation (∑F = ma + (dm/dt) * v) is a form of Newton's second law applied to a variable-mass system, such as a rocket that ejects fuel.

$$ \sum \vec{F} = m \vec{a} + \frac{dm}{dt} \vec{v} $$

∑F: The sum of the external forces acting on the system (thrust generated by the ejection of gases + gravity + air resistance).
ma: The product of the system's mass and its acceleration.
(dm/dt) * v: The term that represents the variation of momentum (p=mv) due to the variation of the system's mass.

This equation accounts for a variation in mass. It is particularly useful for describing the motion of a rocket, as its mass continuously decreases by burning fuel and expelling gases. When fuel is ejected, it carries away a certain amount of momentum. For the rocket to take off, it must gain momentum in the opposite direction to the fuel ejection. This transfer of momentum allows the rocket to accelerate and take off.

In other words, thrust generates a continuous increase in speed, which in turn leads to an increase in momentum, i.e., (dm/dt) * v, despite the decrease in mass. The increase in speed thus leads to an increase in momentum, as �� is proportional to ��. And the decrease in mass leads to a decrease in momentum (dm/dt) * v, as �� is proportional to m.

The key to understanding why momentum increases lies in the fact that the rocket's speed increases faster than its mass decreases.

Detail of the Dynamics Calculation

By differentiating this expression with respect to time, we obtain: $$ \frac{d\vec{p}}{dt} = \frac{d}{dt} (m \vec{v}) $$ Applying the product rule: $$ \frac{d\vec{p}}{dt} = m \frac{d\vec{v}}{dt} + \frac{dm}{dt} \vec{v} $$ Since acceleration is defined by $ \vec{a} = \frac{d\vec{v}}{dt} $, this equation becomes: $$ \sum \vec{F} = m \vec{a} + \frac{dm}{dt} \vec{v} $$

∑F: sum of the external forces acting on the system, expressed in newtons (N),
ma: mass of the object (in kg) times the acceleration of the center of mass (in m/s²),
dm/dt: variation of the system's mass per unit time (in kg/s),
v: instantaneous velocity of the center of mass (in m/s).

Real-World Examples

The dynamics equation shows that when mass varies, an additional term $ \frac{dm}{dt} \vec{v} $ must be considered. This term is crucial for explaining:

The thrust of a rocket in flight, where the mass of the craft decreases due to the ejection of propellant gas.
A turbojet is a dynamic variable-mass system: the mass of the incoming fluid differs from the mass exiting after combustion and acceleration. This difference is essential for generating the thrust needed for aircraft propulsion.
The propulsion of jellyfish that expel water to move through a fluid by changing their instantaneous mass. The jellyfish loses mass because it expels the water contained in its bell.
The mass of a hot air balloon changes due to the variation in the temperature of the air it contains. This phenomenon causes air exchanges with the outside, altering the total mass and affecting flight dynamics.
The ejection of matter by an exploding star (supernova), where the variation in mass modifies the trajectory and speed of the outer layers.
Icebergs breaking off from a glacier, where the loss of mass affects the stability and movement of the whole.
…

Momentum dynamics is a more general reformulation of Newton's law, essential for understanding systems where mass varies. It plays a key role in space mechanics, aerodynamics, and fluid mechanics.