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 $\beta=V / c$