Four Chapter Two

The Transfer of energy in Simple Collisions

• Now that we have discussed Kinetic and Potential energy and how it is transformed when interactions occur, we can go back and discuss what role energy plays in collisions.
• A moving object which collides with a stationary object, can cause the stationary object to move
  • i.e. Kinetic energy is transferred from one to the other.
• Consider the collision we analysed before when talking about momentum in section "Two Chapter One."
• The velocities we determined before for these two objecs are:

• From these results, you can see that the kinetic energy was transfered from the incident puck to the stationary puck.
• We can also see that kinetic energy was not conserved.
• One can ask, why did these results occur?
  • The answer lies in the fact that some energy was transformed into heat and sound.
• There are three types of collisions
  • Elastic collision - collisons in which
    • Total energy is conserved
    • Total momentum is conserved
    • Total kinetic energy is conserved
  • Inelastic collision - collisions in which
    • Total energy is conserved
    • Total momentum is conserved
    • Total kinetic energy is not conserved - some portion of kinetic energy is converted to other forms of energy or does work to deform the objects.
  • Completely inelastic collision - collisions in which
    • Total energy is conserved
    • Total momentum is conserved
    • Total kinetic energy is completely lost. This means that after collision the kinetic energy equals zero. For this to happen, there must be no motion present after collision.

Elastic Collisions

• Consider the collision of a super ball with another super ball (stationary) as shown

• During contact, the balls become compressed and behave according to Hooke's law

• Xmax represents the maximum compressions during collision.
• Before contact, ball two has no kinetic energy (Ek₂) .

Ek₂ = (1/2)m₂V₂² = 0 J

• During contact, ball one experiences a force backwards to slow it down, while ball 2 experiences a forward force to speed it up.
• This force also compresses the balls
  • During compression, a force is applied through a distance to do work (shaded area on the graph).
  • This work results in a build up of elastic potential energy (E_e) in the balls equal to the shaded area on the graph
  • Therefore total energy (E_T) during contact is given by

E_T = Ek₁ + Ek₂ + E_e

• As the collision proceeds, the kinetic energy of ball one will continually decrease from the moment of first contact, to when the balls are separated (i.e. V₁ decreases).
• As the kinetic energy of ball one decreases, the kinetic energy of ball two will be continually increasing while the balls are in contact (i.e. V₂ increases).
• Elastic potential energy increases until maximum compressions occurs at Xmax, and then the compression will decrease while the elastic potential energy decreases back to zero once the contact has ended.
• In summary:
  • Prior to maximum compression V₁ > V₂
  • At maximum compression V₁ = V₂
• After maximum compression V₁ < V₂
• For elastic collisions, the total energy remains constant, but during contact, Total kinetic energy is lost by conversion to elastic potentials energy, which converts back to kinetic energy after maximum compression so that:

Ek_{before contact} = Ek_{after contact}

Elastic and Inelastic Collisions

• The difference between elastic and inelastic collisions is whether the energy stored as potential energy during contact, is retrieved as kinetic energy after contact.
• Retrieval depends upon the mechanisms operating during contact.
  • For example, if during conpression, an object is permanently deformed and cannot return to its original shape, then work done to cause the deformation uses up kinetic energy and then:

Ek_{Total before} > Ek_{Total after}

• If work is done to use up kinetic energy in deforming, or other forms of energy appear during contact, that are not transformed back to kinetic energy after, then kinetic energy is lost, and total energy is then given by

E_T = Ek₁ + Ek₂ + E_e + Q + Light + Sound + .......

• This causes the Force - Distance (compression) graph (F - d) to have a different shape depending upon whether compression is occuring (decreasing distance between objects) or expansion is occuring (increasing distance between objects).
• Conditional difference of the Force - Distance graph for elastic and inelastic collisions:
  1. If the F - d graph has the same shape for decreasing distance as increasing distance, then the collision is elastic

1. If the F - d graph has a different shape for decreasing distance than increasing distance, then the collision is inelastic.

• For inelastic collisions, if we superimpose these graphs, then the area between the curves represents the lost kinetic energy.

December 17, 2013