Common mistakes of adding up vectors of acceleration*

A mathematical example of incorrect management of acceleration vectors is the Special Relativity Assumption that if a train rider drops a stone off his carriage while moving in a uniform motion, he will see it moving in a straight line. However, the better expectation, mathematically, is that he will see it moving in a parabola, just like any other outside observer. When either an observer or a projectile, subjected to at least one different acceleration, the observed path of projectile follows a mathematical equation of the second degree, which represent a parabolic curve line on an ( X-Y ) coordinates.
Similarly, calculating energy of a given system as a scaler is based on the incorrect assumption that all acceleration vectors of any motion, including a net force and equilibrium can be summed up into one vector and if such system is the universe, then we end up incorrectly speaking of a cold expansion and ignoring energies used in the non linear accelerations.