Momentum

Linear Momentum

  • Defined as product of mass (m) and vector velocity (v).

  • Momentum (p) is a vector quantity; direction is important for problem-solving.

  • Unit of momentum: kilogram meter per second (kg m/s).

Revisiting Newton's Second Law

  • Original: Net force (F) = mass (m)

    acceleration (a).

  • Acceleration can be expressed as rate of change of velocity.

  • If mass changes, rewrite Newton's second law in terms of momentum: net force is equal to the rate of change of momentum.

  • This accounts for scenarios like a rocket losing fuel.

Impulse

  • Defined as change in momentum, which is equal to the force multiplied by the time duration over which the force acts.

  • Impulse (J) is also a vector quantity:

  • Longer time of force application (delta t) results in smaller net force (F).

Examples

  • Eggs Dropping:

    • Both eggs have the same change in momentum when dropped from the same height.

    • Impact with a pillow (longer delta t) results in smaller net force compared to hitting concrete, reducing damage.

  • Bouncing Ball:

    • A ball with mass (M) hits the floor with speed (V

      ₀);

    • After bouncing, speed is (1/2)V

      ₀. Time of contact is T_c.

    • Change in momentum: calculated by considering the initial and final velocities.

    • Net force during collision: calculated as the change in momentum divided by the time of contact, directed upwards (positive direction).

Impulse and Non-Constant Forces

  • Definition of Impulse (J):

    • Change in momentum

    • Vector quantity

  • Impulse-Momentum Relation:

    • Describes the relationship between impulse and change in momentum.

  • Variable Forces:

    • Impulse can also be calculated for variable forces.

    • Represents the total impulse from forces that change over time.

  • Key Distinctions Between Impulse and Work:

    • Impulse: Concerned with force as a function of time.

    • Work (W): Represents change in kinetic energy.

      • Involves forces that vary with position.

  • Application Example:

    • For a force that changes over a time interval:

      • Change in momentum can be expressed as an integral of the force over time.

      • Integral evaluated from initial to final time.