# MOMENTUM and COLLISIONS

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

• Newton’s 2nd law of motion for a single particle can be written,

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.

• For a system of particles we have just seen that   .  This expression can be written

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

• Suppose now that Fext = 0, that is there are no external forces acting on the system.  In this case,   which means that P is constant.
• Conservation of Momentum

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
• Choose the system such that there are external forces only in the y direction; then momentum is conserved in x and z (e.g. billiard ball collisions).
• Choose the system such that the external force is much smaller than the internal forces involved during a collision.  In this case momentum will not be exactly conserved, but applying momentum conservation will be a good approximation.

• Application of Momentum Conservation to Collisions

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
and another one from Bill Nye

• Impulse and Collisions

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.

•   Food for thought…
• We have seen that velocity is “relative”, the value assigned to a particle depends on the reference frame of the observer.  Therefore, momentum (and kinetic energy) is also dependent on the observer’s reference frame.  Different observers of the same system may measure different values for P and pi, but due to momentum conservation each will find his/her value of the total momentum unchanged.
• Conservation of energy, momentum and mass are extremely important in describing nature.  In the realm of large velocities (approaching the speed of light) Newtonian mechanics is no longer  valid – we must use Einstein’s theory of Relativity  To describe the interactions of matter at very small distances we must use the theory of Quantum Mechanics.  However, energy, momentum and mass conservation remain valid under Relativity and Quantum Mechanics.  These conservation laws are certainly more fundamental than Newton’s Laws of motion.

"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