Deriving Relativistic Mechanics and a Physical Frequency Related to Energy from a Linear Algebraic Model of a Bouncing Particle

By assuming that a particle acts as a system moving back and forth while moving on average in one direction, one can obtain Einstein’s 1905 energy momentum equation E*E = p*p + m*m. The bouncing motion is modeled using linear algebra with velocity V and acceleration dV\dt  matrices emerging. It is found that if a bouncing particle with no translation, i.e. no velocity v,  is represented by (1,1), then accelerating the particle leads to a state of |v>=a(1,1) + b(1,-1), with a and b being coefficients related to v. Both (1,1) and (1,-1) represent bouncing particles, but have different phases.  It is shown that V|v> = v|v> + sqrt(1-v*v) |v perpendicular>, thus leading to the energy momentum equation. The energy momentum equation is also obtained using  the square of the velocity matrix. It is also argued that  (1,1) and (1,-1) can interfere and are analogous to sin(wt) and cos(wt). It is further shown that w, the frequency , is the energy term in the energy momentum equation. Hence, the frequency which is the energy, is related to a physical frequency in this model.