General Physics (PHYS 101)

Linear Momentum and Collisions

Content Overview
  • Linear Momentum
  • Impulse
  • Conservation of Linear Momentum
  • Elastic and Inelastic Collisions
  • Center of Mass
  • Jet Propulsion and Rockets

Linear Momentum

  • Definition of Linear Momentum
    • Linear momentum ( extbf{p}) of an object is defined as the product of its mass (m) and velocity (v).
    • Mathematically, it is expressed as:
      extbfp=mimesextbfvextbf{p} = m imes extbf{v}
    • Note: Momentum is a vector quantity, possessing both magnitude and direction.
    • SI Unit of Momentum: kg • m/s (no special name).
Sample Problem

  • Scenario: A 3000 kg elephant chases a 1 kg squirrel with given velocities.

    • Elephant's Momentum Calculation:

    • Given: mass = 3000 kg, velocity = 5 m/s (west)

    • extpextelephant=mimesv=(3000extkg)imes(5extm/s)=15000extkgm/sextwestext{p}_{ ext{elephant}} = m imes v = (3000 ext{ kg}) imes (5 ext{ m/s}) = 15000 ext{ kg m/s} ext{ west}

    • Squirrel's Momentum Calculation:

    • Given: mass = 1 kg, velocity = 7 m/s (west)

    • extpextsquirrel=mimesv=(1extkg)imes(7extm/s)=7extkgm/sextwestext{p}_{ ext{squirrel}} = m imes v = (1 ext{ kg}) imes (7 ext{ m/s}) = 7 ext{ kg m/s} ext{ west}

Total Momentum for a System
  • In a system of multiple objects, the total momentum is computed as the vector sum of the momenta of each individual object.

Change in Momentum

  • Definition: The change in momentum of an object is defined as the difference in its momentum vectors before and after an event.
Net Force and Momentum Change
  • If an object's momentum changes, it indicates that a net force has acted on it. The relation is given by:
    • extNetForce=extChangeinMomentumextChangeinTimeext{Net Force} = \frac{ ext{Change in Momentum}}{ ext{Change in Time}}

Impulse

  • Definition: Impulse is the product of the average net external force ( extbf{F}_{net}) and the time interval ( ext{Δt}) during which it acts, signifying a change in momentum:
    • extImpulse=extbfFnetimesextΔt=extΔpext{Impulse} = extbf{F}_{net} imes ext{Δt} = ext{Δp}
Impulse-Momentum Theorem
  • This theorem states that impulse is equal to the change in momentum of an object. Thus, it is crucial for understanding dynamic interactions in collisions.
Characteristics of Impulse
  • The effect of a force on an object is profoundly influenced not only by the force's magnitude but also by the duration of its action.
  • Intuitive Understanding: When a moving object halts, its impulse reflects solely its change in momentum.
  • Large forces applied briefly (e.g., a tennis racket hitting a ball) can effect significant momentum change. Conversely, smaller forces can achieve the same result but require a longer duration.
Practical Examples of Impulse
  • Landing Softly: Bending the knees during a jump helps to reduce the force felt upon landing; a softer catch (moving hands) minimizes pain compared to a hard catch (fixed hands).
  • Safety Mechanisms: Airbags and crumple zones in vehicles extend the duration of force application during collisions, thus reducing injury risks by spreading impulse over time.

Conservation of Linear Momentum

  • Key Principle: In a closed system where no net external forces are acting, total momentum remains constant.
    • This principle underlines the law of conservation of momentum. Internal forces may alter individual momenta within the system but do not affect the overall momentum.
Collisions and Momentum Conservation
  • During collisions, which occur rapidly enough for external forces to be negligible, momentum is conserved. This foundational principle allows for analyses of both elastic and inelastic collisions.

Elastic and Inelastic Collisions

  • Elastic Collision: Both total kinetic energy and momentum are conserved.
  • Inelastic Collision: Only momentum is conserved; kinetic energy is not conserved.
  • Completely Inelastic Collision: The objects stick together post-collision, demonstrating maximum energy dissipation in the form of heat or deformation.
Conservation in Collisions
  • In general, the total kinetic energy fraction remaining after a completely inelastic collision is defined, while for elastic collisions, both kinetic energy and momentum are conserved.
Worked Example of Collision
  • Scenario: Two gliders of differing masses on a frictionless air track collide.
  • Analysis includes determining final velocities post-collision, emphasizing that both gliders experience equal and opposite impulses due to Newton's Third Law, affecting momentum changes accordingly.

Jet Propulsion and Rockets

  • Principle: Jet propulsion is illustrated using the balloon example; as air escapes, it propels the balloon in the opposite direction.
  • This exemplifies the conservation of momentum, where the force exerted by escaping air induces an equal reaction force on the balloon.
  • Rocket Propulsion: The operational principle of rockets mirrors this, with engines exerting downward force on exhaust gases which push upward on the rocket.
Impulse in Rocketry
  • The formula relating rocket and gas forces incorporates mass, acceleration, and time:
    • extForceimesextTime=extMassimesextAccelerationimesextTime=extMassimesextChangeinVelocityext{Force} imes ext{Time} = ext{Mass} imes ext{Acceleration} imes ext{Time} = ext{Mass} imes ext{Change in Velocity}
  • Therefore, the overall change in momentum of both the rocket and the exhaust is effectively zero.

Summary

  • Momentum: Defined as the product of mass and velocity; for a system, the total momentum is the vector sum of its components.
  • Impulse-Momentum Theorem: In the absence of external forces, momentum is conserved. Momentum conservation occurs in collisions, with kinetic energy conserved in elastic collisions.
Thank You
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