Collision Processes

Air molecules collide with the leaves of a tree. The state of motion of each air molecule is changed during these collisions. The state of motion of each leaf is also changed during these collisions.

An asteroid narrowly missing the Earth exhibits a change in its state of motion. Its orbit about the sun is changed from perhaps an elliptical orbit to a more circular orbit. The orbit of the Earth is also changed. These changes may be very small, but they exist.

When you hit a baseball with a bat, the abrupt change in the motion of the ball is obvious. The bat applies a large force to the ball for a short period of time causing the ball to change its state of motion. The state of motion of the bat is also changed. You can feel this change if you are holding the bat.

Collision processes are very complex. We try to abstract essential aspects of these processes in the form of conservation laws. Conservation laws allow us to focus our attention on measurable quantities. Examining these measurable quantities will allow us to make predictions about complicated processes.

Consider the following self-evident statements:

The molecule collides with the leaf while the leaf collides with the molecule.
The asteroid collides with the Earth while the Earth collides with the asteroid.
The ball collides with the bat while the bat collides with the ball.

Convince yourself of the "truth" of:

Newton's Third Law of dynamics

• You can't touch without being touched.
• For every action there is an equal and opposite reaction.
• F_1,2 = - F_2,1

Next, you need to convince yourself that the following statement is true.

During each instant of time , the force applied (to the bat by the ball) is equal and opposite to the force applied (to the ball by the bat).

The Mathematics

Typically collision processes happen so fast that the details are difficult to determine. Furthermore, usually, we are only interested in being able to predict the outcome of a collision. Or, if we know the outcome of a collision, we will want to be able to infer the conditions prior to the collision. Newton's third law of dynamics provides us with the mathematical tools to do these things.

Consider a situation in which two particles (label them #₁ and #₂) are moving toward one another. Further, imagine these two particles to be moving in a vacuum and in a region far from any other objects. With these restrictions we need only concern ourselves with forces acting between the two objects.

#₁ has a mass, m₁ and is moving with a velocity, v₁.

#₂ has a mass, m₂ and is moving with a velocity, v₂.

When the two bodies are close together, #₁ will feel a force F_1,2 acting upon it, while #₂ will feel a force F_2,1 acting upon it.

The only requirement for a collision to occur is that the relative velocity vector, v_rel = (v₁ - v₂), points from #₁ toward #₂.
Draw a few pictures to convince yourself that this "initial condition" statement is true.

With

1. this initial condition,
2. Newton's third law, and
3. Newton's second law (F = ma)

Apply (3.) to #₁ during the time (Δt) of the interaction : F_1,2 = m₁(Δv₁/Δt).
Apply (3.) to #₂ during the time (Δt) of the interaction : F_2,1 = m₂(Δv₂/Δt).

We will find it helpful to write Δv₁ as (v₃ - v₁), and Δv₂ as (v₄ - v₂).
With this notation we are saying that during the interaction #₁ experiences a change in velocity from v₁ to v₃, while during the interaction #₂ experiences a change in velocity from v₂ to v₄.

Apply (2.) Newton's 3^rd law ( F_1,2 = - F_2,1 ) to the above equations,
then do a little algebra to produce the equation :       m₁v₃ + m₂v₄ = m₁v₁ + m₂v₂.

If we identify the quantity mv as a momentum, the equation tells us that (the sum of the momenta after the collision) equals (the sum of the momenta before the collision).

When we discuss collisions along a straight line we can drop the vector notation to write the conservation of momentum equation as :

      m₁v₃ + m₂v₄ = m₁v₁ + m₂v₂,

and let the signs (+/-) on the velocity values indicate direction.

Kinetic Energy Considerations

While it is always true that momentum is conserved during a collision, the total kinetic energy after the collision may be quite different from the total kinetic energy before the collision.

If it happens that the total kinetic energy after the collision is equal to the total kinetic energy before the collision, the collision is said to be an elastic collision. If this is the case we write the equation :

      (¹/₂)m₁(v₃)² + (¹/₂)m₂(v₄)² = (¹/₂)m₁(v₁)² + (¹/₂)m₂(v₂)².

Given that the conservation of momentum equation is always true, show that the conservation of kinetic equation reduces to :

      v₄ - v₃ = v₁ - v₂.

If a collision is an inelastic collision the two bodies stick together and we write

      v₄ - v₃ = 0.

If a collision is a semi-elastic collision the two bodies separate after the collision such that

      v₄ - v₃ < v₁ - v₂.

We may summarize the equations for the collision of two objects along a straight line thus:

1. m₁v₃ + m₂v₄ = m₁v₁ + m₂v₂

2. ( v₄ - v₃ ) = e ( v₁ - v₂ )

   e = 0 , Inelastic Collision

   e = 1 , Elastic Collision

   0 < e < 1 , Semi-Elastic Collision

The symbol "e" is given the name "coefficient of elasticity". If e > 1 , then the collision is an explosion or a super-elastic collision.

Examples

Examine the three major classes of collisions

• Inelastic Collision
• Elastic Collision
• Semi-Elastic Collision

and summarize the various aspects of these collision processes.