Momentum Study Notes

Chapter 6: Momentum

6.2 Impulse

  • Definition of Impulse:

    • Impulse is defined as the product of force and the time interval during which the force acts.
    • Shorthand notation: Impulse = Ft

  • Concept of Momentum Change:

    • If the momentum of an object changes, it implies that either the mass or the velocity has changed, or both.
    • In most cases, mass remains unchanged, indicating that any change in momentum is due to a change in velocity which involves acceleration.

  • Force and Acceleration:

    • The change in velocity in a given time interval depends on the net force acting on the object.
    • Greater net force leads to greater change in velocity, and thus greater change in momentum.

  • Importance of Time:

    • The duration for which a force is applied (time interval) is crucial in determining the change in momentum.
    • Example:
    • A brief force applied to a stalled automobile produces a small change in momentum.
    • The same force applied over a longer time results in a greater change of momentum due to sustained force application.

  • Summary of Key Points:

    • Both force and time interval are vital for changing momentum.

6.3 Impulse-Momentum Relationship

  • Impulse-Momentum Relationship:

    • If a greater impulse is exerted on an object, it will experience a greater change in momentum.
    • Stated as: Impulse = change in momentum.
    • Formula:
      extImpulse(Ft)=riangle(mv)ext{Impulse (Ft) = } riangle (mv)
    • The delta symbol (Δ) represents a change or difference, typically used in relation to momentum changes.

  • Derivation of the Impulse-Momentum Relationship:

    • Derived by rearranging Newton's second law (F = ma) while expressing acceleration and time factors:
    • Acceleration: a=rianglevriangleta = \frac{ riangle v}{ riangle t}
    • Substituting gives:
      Fm=rianglevrianglet\frac{F}{m} = \frac{ riangle v}{ riangle t}
    • Rearranging leads to:
      Frianglet=riangle(mv)F riangle t = riangle (mv)

  • Impulse Examples:

    • Impulse can relate to increasing momentum, decreasing momentum over a longer time, and decreasing momentum over a shorter time.
Case 1: Increasing Momentum

  • Application of Force:

    • To increase momentum effectively, apply the maximum force possible for the longest time possible.
    • Examples include golfers and baseball players who apply maximum force while following through with their swings to increase contact time with the bat or club.

  • Impulse Variation:

    • The forces acting during the impulse may vary from moment to moment.
    • Example of varying force: A golf club striking a ball initially exerts zero force until contact is made.
Checkpoint Questions for Case 1:
  1. Which cannon imparts greater speed to a cannonball: a long-barreled or short-barreled cannon?
    • Answer: Long-barreled cannon produces greater speed due to the longer time over which force is exerted, leading to greater impulse.
  2. Between a long and short drinking straw, which moves a pea farther with the same breath?
    • Answer: A pea shot from a long straw will travel farther for the same reasons as the cannon; it generates greater impulse and speed.
Case 2: Decreasing Momentum Over a Long Time

  • Safety in Collisions:

    • Choosing to hit a soft object (e.g., haystack) instead of a hard one (e.g., concrete wall) extends the time over which momentum is brought to zero, thus reducing the force experienced.
    • Example: Landing on a haystack results in extended stopping time and reduced force, whereas a sudden stop like hitting a wall produces high force and deceleration leading to injury.

  • Time Extension Effect:

    • Extending the time of contact leads to a substantial reduction in force.
    • Example: If contact time is extended 100 times, the force reduces to one-hundredth of the original force.

  • Practical Applications:

    • Designs like padded dashboards and airbags leverage this principle to save lives in automobile crashes by extending the time over which deceleration occurs.
Checkpoint Questions for Case 2:
  1. What physics accounts for the importance of safety nets for circus acrobats?
    • Answer: The impulse-momentum relationship extends the stopping time, thereby reducing the stopping force when landing.
  2. Is the impulse halting an acrobat's fall the same for a floor versus a safety net?
    • Answer: Yes, the impulse is the same but the time of contact is increased with a net, leading to a softer landing force.
Case 3: Decreasing Momentum Over a Short Time

  • High-Speed Impacts:

    • Reducing contact time results in increased force.
    • In boxing, if a fighter moves into a punch, the time of impact decreases and thus the force increases, potentially resulting in greater damage.

  • Karate and High Impacts:

    • A karate expert can split bricks by delivering a swift punch; the quick impact causes a large force due to the short time of contact.
Checkpoint Questions for Case 3:
  1. How much will the force of impact decrease if the boxer increases impact duration to three times?
    • Answer: Force will decrease to a third of its original value.
  2. What happens if the impact time is halved?
    • Answer: The force of impact will double.
  3. Are there contradictions in impacts sustained by boxers versus karate experts?
    • Answer: No contradiction; each scenario seeks a different outcome: reduced force for boxers via longer duration vs. increased force for karate practitioners via shorter duration.
  • Final Notes:
    • Both longer and shorter impacts can lead to the same impulse without conflicting physics concepts.