The study of momentum and impulse is useful to solve problems of collisions and explosions.

Linear Momentum

Important quantities are:

Linear Momentum \((\vec{P}\) )
The product of the mass \(m\) of a particle by its velocity \(\vec{v}\) is the momentum, mathematically: $$ \vec{P} = m \vec{v}.$$ As clarified by the equation, the direction of the momentum is the same of the speed, and its \(IS\) unit is \([P] = kg \frac{m}{s}\).
Impulse \((\vec{I})\)
The average force \(\vec{F}_a\) times the actuation time \(\Delta t\) gives the impulse, in mathematical form: $$\vec{I} = \vec{F}_a \Delta t,$$ where the linear momentum and direction of the force are the same \(\vec{F}_a\) .

In a graphic of force versus time, the area under the curve is numerically equal to the impulse of the force in the relevant time interval.

Theorem of Momentum

The total impulse that receives an object determines its variation amount of movement: $$\vec{I} = \Delta \vec{P}$$ or $$\vec{F}_a \Delta t = m \vec{v} - m \vec{v}_0.$$ This theorem is applicable if:

  • The \(\vec{F}_a\) is much greater than any other force present in the system of interest.
  • The \(\Delta t\) is small, so that the displacement is negligible during the collision.

Theorem of Conservation of Momentum

When the sum of all external forces acting on a system is zero, the total momentum of the system remains unchanged, i.e., the momentum is constant. More precisely, we can formulate this theorem as: for some amount of motion \(\vec{P}_A\) at time \(A\) and a quantity \(\vec{P}_B\) at a later time \(B\), we have: $$\vec{P}_A = \vec{P}_B$$ if $$\sum_i \vec{F}_{i} = \vec{O}.$$

Forces on Particle Systems

The system of interest could be considered as a particle system when consisting of more than one moving part. In this case the following quantities are important:

Internal Forces \((\vec{F}_{int})\)
Forces exchanged between the bodies of the system itself are called internal forces.
External Forces \((\vec{F}_{ext})\)
Forces exchanged between the system and bodies outside the system are external.
Net Force \((\vec{F}_{net})\)
The resulting force or net force on a system of particles is given by: $$F_{net} = \sum \vec{F}_{int} + \sum \vec{F}_{ext}.$$ However, by Newton's third law, the law of action and reaction: $$ \sum \vec{F}_{int} = \vec{0}.$$ This means that the internal forces do not contribute to the displacement of the system as a whole, but can contribute to the displacement of its parts.
Isolated System
In an isolated system, the sum of the external forces is zero, i.e., $$ \sum \vec{F}_{ext} = \vec{0}$$ Examples of systems that may be considered mechanically isolated:
  • When no external force acts on the system. For example, a spaceship in outer space far from any celestial body.
  • When external forces are negligible in relation to the internal. Examples: shocks, explosions, shooting guns, etc.
  • When the external forces acting on the system are neutralized.