In special relativity, the equation you mentioned, E = sqrt((mc^2)^2 + (PC)^2), is the relativistic energy-momentum relation, also known as the mass-energy equivalence relation. It relates the energy (E) and momentum (P) of a particle with its rest mass (m) and the speed of light (c).
While the momentum of a particle can be expressed as P = mv, where m is the rest mass of the particle and v is its velocity, substituting mv into the equation for the relativistic energy would not give the correct result. This is because the relativistic energy-momentum relation takes into account the relativistic effects that arise at speeds close to the speed of light.
In special relativity, the momentum of a particle is given by:
P = γmv
where γ is the Lorentz factor, defined as γ = 1 / sqrt(1 - (v^2/c^2)). The Lorentz factor accounts for time dilation and length contraction effects that occur as an object approaches the speed of light.
By using the correct expression for momentum in the relativistic energy-momentum relation, the equation becomes:
E = sqrt((mc^2)^2 + (γmv)^2)
This equation properly incorporates the relativistic effects and provides an accurate calculation of the energy of a particle. Simply substituting mv for P and inserting the velocity would neglect the relativistic corrections, leading to incorrect results.