Surfing on moving potential waves

The solution to a problem involving the calculation of the kinetic energy of a particle moving in a potential well with the use of Galilean transformation, Lorentz transformation and special relativity is discussed. The kinetic energy of a particle falling in a potential well is higher than the potential energy stipulated of the potential well's height. Energy of the particles in quantum wells are essential in particle accelerators, cosmic ray travels and generation of solar winds and aurora borealis.