MOMENTUM and COLLISIONS



"What one man can invent another can discover"
Arthur Conan Doyle


 

 

 

where we have assumed that the mass of the particle is independent of time and we define the momentum of the particle, p = mv.  Momentum is a clearly a vector quantity, with units, kg.m/s  (SI) or slug.ft/s  (British).

 Note that writing the 2nd law as force equal to the rate of change of momentum is the form in which Newton developed the law.

 

 

 

where P is the vector sum of all the individual particle momenta in the system.


Important basic principle of Science

"Total momentum of a system remains constant,
when the net external force acting on the system is zero"

Within the system objects may collide with each other, thus exerting forces on each other.  However, Newton's 3rd law states that these forces are equal in magnitude, but opposite in direction, therefore there is no net force.  The system conserves momentum.  In order to ensure momentum conservation we must choose our "system" such that the net force is zero, in which case;

"Total momentum of system before collision = Total momentum of system after collision"

Note that since momentum is a vector quantity, this equation is actually three scalar equations, one for momentum along each of the three Cartesian axes, x,y,z.

 The condition that the net external force on the system be zero appears to make momentum conservation less basic that energy conservation (since there is no such condition for energy).  However, if we define the Universe as our system, then all forces are internal and universal momentum conservation is guaranteed.


 On a more practical note, since we do not want to consider the whole Universe every time we apply momentum conservation, we can consider external forces zero in the following two situations

 

Given that we can assume momentum is conserved there are two classes of collisions

o       ELASTIC:      Kinetic energy is conserved, KEinitial = KEfinal

o       INELASTIC:  Kinetic energy is not conserved.

§         If the maximum kinetic energy is lost, consistent with momentum conservation, the collision is called Completely Inelastic.  This situation can be achieved when two objects stick together after colliding.

§         In inelastic collisions the “lost” kinetic energy is converted into some other form of energy – heat, sound, elastic etc.

§         The total energy in a system is always conserved, but in collisions we are usually only able to easily measure kinetic energies, which means we can apply (kinetic) energy conservation only to elastic collisions.

  Remember, momentum is a vector quantity.  Thus, in three dimensions, application of conservation of momentum will lead to 3 equations relating the components of momentum before and after a collision.

Momentum conservation video example

 

Using the alternative formulation of the second law,   , we can write

 

 

 

The impulse is defined here for a single object, during a 2 body collision the impulse on the first object will be equal and opposite to that on the second.  Consideration of the impulse is most useful when the force during a collision is not constant.

 


 


"To mark the opening of a sports hall for juvenile offenders the Prime Minister up rooted a tree."

Ronnie Barker


 

Dr. C. L. Davis
Physics Department
University of
Louisville
email: c.l.davis@louisville.edu