Collisions and Conservation of Momentum

Our current understanding of collisions traces its origins back to the studies of John Wallis and Christopher Wren (and upon who Newton based his work).  Most textbooks will break collisions into two types, elastic and inelastic.  We might define an elastic collision as one in which the two colliding objects bounce off of each other, and an inelastic collision as one in which the two colliding objects stick together. Wallis developed the theory for inelastic collisions and Wren developed the theory for elastic collisions.

I would prefer to expand this into four categories:

Explosions:  Two objects which are stuck together fly apart due to an internal force (such as a spring, magnetism, chemical explosion, nuclear explosion, etc.)

Perfectly Elastic Collision: Two objects which when colliding bounce off of each other and never actually touch.  The force which repels them is a force that acts at a distance such as magnetism or the electric force.  Since they never touch, there is no light, heat or sound generated, and there is no deformation of the objects involved in the collision.

Elastic Collision (non-perfect): Two objects which when colliding bounce off of each and actually come into contact briefly.    Since they do touch, there is may be light, heat or sound generated, and there might be some deformation of the objects involved in the collision.

Inelastic Collision: Two objects when colliding stick together after the collision.  In this process, there may be heat or sound generated.  There very well likely will be some deformation of the objects involved in the collision.

Newton (and Descartes) based his work in refining the theory of Conservation of Momentum on the work of Wallis and Wren.  However, there was a competing theory of collisions that persisted for hundreds of years called Vis Viva.  Leibniz developed the theory of Vis Viva looked at other conservation factors in collisions.  Newton and Leibniz were often at odds.  Both independently came up with calculus, but with very different notations.  I will return to this in a later posting.

Conservation of Momentum and Newton’s 3rd Law

We defined momentum as inertia in motion, or

p = mv

When we examine a system (which may two two objects, or multiple objects), the Law of Conservation of Momentum states that the momentum of a system is conserved IF there is no external force.   An external force is a force that is NOT part of the system, as opposed to an internal force.  In the example of two exploding carts, the spring would be an internal force.  With two carts that collide, the magnets would be an internal force.  You pushing the carts would be an external force.   The origins of this idea can be found in Newton’s 3rd Law of Motion.  Although we might also say that Newton’s 3rd law results from the idea of Conservation of Momentum (as developed by Wallis).  We might also think of the momentum of a system as the Net Momentum, or the sum of the momenta of all the parts. Oh, and yes, the plural of momentum is momenta.

Let us begin by examining the case of a one-dimensional explosion.

Newton’s 3rd Law states that for every action there is an equal and opposite reaction.  With two objects exploding, object 1 exerts a force on object 2.  Object 2 exerts an equal and opposite force back on object 1.

F12=-F21

This explosion lasts for a duration of time we will call Δt.  If we multiply both sides of this equation by this time, then we have impulse.

F12Δt=-F21Δt

In turn, with the momentum-impulse relationship, we can say the change in momentum of object one is equal and opposite to the change of momentum of object 2

Δp1=-Δp2

Another way to look at this is we could say whatever momentum object 2 loses, object 1 gains.  Now using our definition of momentum we can say

Δm1v1 =-Δm2v2

So in the case of an explosion, although the two objects will have equal and opposite momentum, their velocities will be proportional to the masses. If the original velocity of both pieces in the explosion is zero, then our equation becomes

m1v1f =-m2v2f

where the subscript f denotes final or after the collision.

Elastic Collisions

Whether it is elastic or perfectly elastic, we can model the transfer of momentum in a collision based on the above analysis.  Starting from

Δm1v1 =-Δm2v2

we can expand with the subscript o for original

m1v1o –  m1v1f = -(m2v2o – m2v2f)

rearranging terms with the events before the collision on the left, and after the collision on the right we get an expression for the Conservation of Momentum in an elastic collision

m1v1o + m2v2o = m1v1f + m2v2f

We can simplify this to

p1o + p2o = p1f + p2f

This can further be restated bring in the idea of Net Momentum as the sum of the momentum of the parts.  In this case, we only have two objects, but it could refer to numerous objects

pNETo = pNETf

Inelastic Collisions

Since in an inelastic collision, the two objects stick together, they have the same final velocity, and our collision equation becomes

m1v1o + m2v2o = (m1+m2)vf

The above work is based on one-dimensional collisions.  When we enter two and three dimensional collisions the analysis becomes complicated.  We will study this next week!