Linear Transformation
Momentum
is a vector quantity which is mathematically defined as p = mv. m is mass and v
is the velocity. When momentum is applied on the pool table, it is a linear
transformation because the balls only move in one unique direction defined by
the momentum of the cue in a straight line.
In terms of forces and assuming constant mass, linear momentum is the rate of
change of the force:
F = m dv/dt = dp/dt
Kinetic Energy
Whenever
objects are in motion (velocity > 0) there is energy produced due to its
movement. This energy is called kinetic energy, the energy of movement. And
mathematically, kinetic energy is defined as:
K = (1/2)mv2
By inspection, we also notice that kinetic energy is the total momentum or
momentum is the rate of change of kinetic energy.
K = int(0, v, mv); p = dK/dv = mv
It is widely known that energy is neither created or destroyed. Mathematically
speaking, this simply means that the initial state of energy in a system is
equal to its final state.
Ei = Ef
This statement also implies that if a ball slows down, its kinetic energy has
been dissipated into some other form however never "gone".