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 [latex]\mathbf{p}[/latex] is defined to be
  [latex]\mathbf{p}=m\mathbf{v},[/latex]

  where [latex]m[/latex] is the mass of the system and [latex]\mathbf{v}[/latex] is its velocity.

• The SI unit for momentum is [latex]\text{kg}·\text{m/s}[/latex].
• 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
  [latex]{\mathbf{F}}_{\text{net}}=\frac{\Delta \mathbf{p}}{\Delta t}\text{,}[/latex]

  [latex]{\mathbf{F}}_{\text{net}}[/latex] is the net external force, [latex]\Delta \mathbf{p}[/latex] is the change in momentum, and [latex]\Delta t[/latex] 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:
  [latex]\Delta \mathbf{p}={\mathbf{F}}_{\text{net}}\Delta t.[/latex]
• Forces are usually not constant over a period of time.

6.3 Conservation of Momentum

• The conservation of momentum principle is written
  [latex]{\mathbf{p}}_{\text{tot}}=\text{constant}[/latex]

  or

  [latex]{\mathbf{\text{p}}}_{\text{tot}}={\mathbf{\text{p}}\prime }_{\text{tot}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(\text{isolated system}\right),[/latex]

  [latex]{\mathbf{p}}_{\text{tot}}[/latex] is the initial total momentum and [latex]{\mathbf{\text{p}}\prime }_{\text{tot}}[/latex] is the total momentum some time later.

• An isolated system is defined to be one for which the net external force is zero [latex]\left({\mathbf{\text{F}}}_{\text{net}}=0\right)\text{.}[/latex]
• 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.