11.3.4 Two Collision Models

No collision is perfectly elastic, although collisions between two very hard objects (metal spheres) or between two springs (such as a collision on an air track) come close. Collisions can be perfectly inelastic, although many real-world inelastic collisions exhibit a small residual bounce. Thus perfectly elastic and perfectly inelastic collisions are models of collisions in which we simplify reality in order to gain understanding without getting bogged down in the messy details of real collisions.

MODEL 11.1 Collisions

For two colliding objects.

  • Represent the objects as elastic objects moving in a straight line.

  • In a perfectly inelastic collision, the objects stick and move together. Kinetic energy is transformed into thermal energy.Mathematically:

    (m1+m2)vfx=m1(vix)1+m2(vix)2
    The figure is the before-and-after pictorial representations of a perfectly inelastic collision where two spherical objects labeled 1 and 2 stick and move together as a single entity after the collision.

  • In a perfectly elastic collision, the objects bounce apart with no loss of energy.

    The figure is the before-and-after pictorial representations of a perfectly elastic collision where two spherical objects labeled 1 and 2 move away from each other after the collision.

    Mathematically:

    • If object 2 is initially at rest, then

      (vfx)1=m1−m2m1+m2(vix)1  (vfx)2=2m1m1+m2(vix)1

    • If both objects are moving, use the Galilean transformation to transform the velocities to a reference frame in which object 2 is at rest.

  • Limitations: Model fails if the collision is not head-on or cannot reasonably be approximated as a “thud” or as a “perfect bounce.”

Workbook Exercise 22