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What is Conservation of Momentum  

 The momentum of an isolated system is a constant. The vector sum of the momenta mv of all the objects of a system cannot be changed by interactions within the system. This puts a strong constraint on the types of motions which can occur in an isolated system. If one part of the system is given a momentum in a given direction, then some other part or parts of the system must simultaneously be given exactly the same momentum in the opposite direction. As far as we can tell, conservation of momentum is an absolute symmetry of nature. That is, we do not know of anything in nature that violates it.

Definition provided by HyperPhysics

Example problem 

       So what that means is if a change in momentum is created an equal change is momentum is created on another object but in the opposite direction.  

       In the diagram if the cannon ball experiences a change in momentum, the gun powder goes off pushing the cannon ball out of the cannon.  The velocity changes from 0 m/s to some other positive value 

       As a result the cannon also experiences a change in momentum, but the change in momentum the cannon experiences is equal to same amount as the cannon ball, but in the opposite direction.  

       Consider it this way, a 5 kg cannon ball is fired out of the cannon with a velocity of 150 m/s 

                   So the change in momentum of the cannon ball is 

     Mass of the cannon ball x change in the velocity of the cannon ball 

                               Mathematically it would look like this 

                              Mass x (velocity final  -  velocity initial)

                            5 kg  x ( 150 m/s  - 0 m/s)   or   750 kg m/s 

Graphic provided by SparkNotes

         The cannon must experience the same change in momentum just in the opposite direction so that we can say mathematically 

Change in momentum of cannon ball  + Change in momentum of the cannon = zero    

 or 

  750 kg m/s + Change in momentum of cannon =zero  

which means that the change in momentum of the cannon is -750 kg m/s

But what this also means that regardless of the situation whether its collision or explosion the total momentum doesn't change.  If you think of it this way, that you change the momentum of the cannon ball with the cannon, this act changes the momentum of the cannon ball changes the momentum of the cannon in the opposite direction.  

so we can say that 

Total initial momentum = Total final Momentum

And that this is always true

 

Types of Collisions 

Elastic and inelastic collisions

Definition by Physics Classroom 

Inelastic Collisions

  Momentum is conserved, because the total momentum of both objects before and after the collision is the same. However, kinetic energy is not conserved. Some of the kinetic energy is converted into sound, heat, and deformation of the objects. A high speed car collision is an inelastic collision. In the above example, if you calculated the momentum of the cars before the collision and added it together, it would be equal to the momentum after the collision when the two cars are stuck together. However, if you calculated the kinetic energy before and after the collision, you would find some of it had been converted to other forms of energy.

Elastic Collision 

  In an elastic collision, both momentum and kinetic energy are conserved. Almost no energy is lost to sound, heat, or deformation. The first rubber ball deforms, but then quickly bounces back to its former shape, and transfers almost all the kinetic energy to the second ball.

Explosion 

  After the explosion, the individual parts of the system (that is often a collection of fragments from the original object) have momentum. If the vector sum of all individual parts of the system could be added together to determine the total momentum after the explosion, then it should be the same as the total momentum before the explosion. Just like in collisions, total system momentum is conserved.