Problem 1.381

A relativistic particle with momentum \(p\) and total energy \(E\) moves along the \(x\) axis of the frame \(K .\) Demonstrate that in the frame \(K^{\prime}\) moving with a constant velocity \(V\) relative to the frame \(K\) in the positive direction of its axis \(x\) the momentum and the total energy of the given particle are defined by the formulas: \[ p_{x}^{\prime}=\frac{p_{x}-E V / c^{2}}{\sqrt{1-\beta^{2}}}, \quad E^{\prime}=\frac{E-p_{x} V}{\sqrt{1-\beta^{2}}} \] where \(\beta=V / c\)