code: 
  import numpy as np
  
  class FORMULAS:
  
      @staticmethod
      def vinf(eninf):
          return np.sqrt(2. * eninf)
  
      @staticmethod
      def vinf_bern(eninf, enthalpy):
          return np.sqrt(2.*(enthalpy*(eninf + 1.) - 1.))
  
      @staticmethod
      def vel(w_lorentz):
          return np.sqrt(1. - 1. / (w_lorentz**2))
  
      @staticmethod
      def get_tau(rho, vel, radius, lrho_b):
  
          rho_b = 10 ** lrho_b
          tau_0 = 0.5 * 2.71828182845904523536 * (radius / vel) * (0.004925794970773136) # in ms
          tau_b = tau_0 * ((rho/rho_b) ** (1.0 / 3.0))
          return tau_b # ms
  
      @staticmethod
      def enthalpy(eps, press, rho):
          return 1 + eps + (press / rho)
  
apis: numpy.sqrt