Section Summary

6.1 Linear Momentum and Force

• Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
• In symbols, linear momentum $\mathbf{p}$ is defined to be
  $\mathbf{p}=m\mathbf{v},$

  where $m$ is the mass of the system and $\mathbf{v}$ is its velocity.

• The SI unit for momentum is $\text{kg}·\text{m/s}$.
• Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
• In symbols, Newton’s second law of motion is defined to be
  ${\mathbf{F}}_{\text{net}}=\frac{\mathrm{\Delta }\mathbf{p}}{\mathrm{\Delta }t}\text{,}$

  ${\mathbf{F}}_{\text{net}}$ is the net external force, $\mathrm{\Delta }\mathbf{p}$ is the change in momentum, and $\mathrm{\Delta }t$ is the change time.

6.2 Impulse

• Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts:
  $\mathrm{\Delta }\mathbf{p}={\mathbf{F}}_{\text{net}}\mathrm{\Delta }t.$
• Forces are usually not constant over a period of time.

6.3 Conservation of Momentum

• The conservation of momentum principle is written
  ${\mathbf{p}}_{\text{tot}}=\text{constant}$

  or

  ${\text{p}}_{\text{tot}}={\text{p}\prime }_{\text{tot}}\phantom{}\phantom{}\left(\text{isolated system}\right),$

  ${\mathbf{p}}_{\text{tot}}$ is the initial total momentum and ${\text{p}\prime }_{\text{tot}}$ is the total momentum some time later.

• An isolated system is defined to be one for which the net external force is zero $({\text{F}}_{\text{net}}=0)\text{.}$
• During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.
• Conservation of momentum applies only when the net external force is zero.
• The conservation of momentum principle is valid when considering systems of particles.

6.4 Elastic Collisions in One Dimension

• An elastic collision is one that conserves internal kinetic energy.
• Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions.

6.5 Inelastic Collisions in One Dimension

• An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).
• A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision.
• Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations.