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• Impulse and quant. of mov.

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 it is: $$\vec{P} = m \vec{v}.$$ 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 it is: $$\vec{I} = \vec{F}_a \Delta t,$$ 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 received by an object determines its variation in the 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, it is: $$\sum \vec{F}_{int} = \vec{0}.$$ 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.

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