Notes on Collisions and Momentum Conservation

Collisions

Definition of Collisions

  • A collision occurs when two objects, free from external forces, strike one another.
  • Examples include:
  • Billiard balls hitting each other
  • A baseball bat hitting a ball
  • A car crashing into another car

Conservation of Momentum

  • Momentum is conserved during collisions.
  • Important to note: Conservation of momentum does not imply conservation of kinetic energy.

Types of Collisions

  • Collisions can be categorized based on kinetic energy:
  • Elastic Collision:
    • Kinetic energy is conserved.
    • The final kinetic energy of the system equals its initial kinetic energy.
  • Inelastic Collision:
    • Kinetic energy is not conserved.
    • Resulting final kinetic energy is less than initial kinetic energy.
    • Completely Inelastic Collision:
    • A special case where colliding objects stick together after the collision.

Example of Collisions

  • The distinction between elastic and inelastic collisions can be illustrated with figures/examples.

Applying Momentum Conservation in Collisions

  • Momentum conservation can be mathematically expressed as:
  • ( m1v1 + m2v2 = (m1 + m2)v_f )
  • Example:
  • If a block of mass m collides with another stationary block of mass m,
    • Mass doubles and speed is halved, translating to a certain final kinetic energy.
    • Some initial kinetic energy is transformed into other forms such as heat and sound.

Real-World Examples of Elastic Collisions

  • Most everyday collisions tend to be inelastic, but examples like billiard balls showcase nearly elastic collisions.
  • Metal balls colliding can also be treated as elastic collisions.

Analyzing Elastic Collisions

  • Both momentum and kinetic energy conservation principles apply to elastic collisions.
  • For two colliding masses (m1 and m2), momentum conservation can be expressed mathematically:
  • Equations derived from momentum and kinetic energy conservation provide solvable variables (final velocities).

Final Velocities in Elastic Collisions

  • The final velocity of cart 1 can vary (positive, negative, or zero) depending on the relative masses of m1 and m2.
  • The final velocity of cart 2 is consistently positive.

Example Problems

  • Example 1:
  • A 1.35-kg block at rest is struck by a 0.0105-kg bullet moving at 715 m/s. Determine the bullet-block system's speed post-collision.
  • Example 2:
  • A 1200-kg car at 2.5 m/s is hit by a 2600-kg truck at 6.2 m/s, causing them to stick together post-collision. Calculate their combined speed immediately after the collision, assuming no external forces act on them.