A rail grinding machine is composed of a rail cart that moves along a rail of a railway via self-propelled locomotion or by being towed. A grinding assembly may simultaneously grind both sides of a vertical web of the rail via respective grinding surfaces that create a ground surface on each side of the vertical web prior to welding one end of one rail to another end of an adjacent rail. The ground surface created by the respective grinding surfaces receives a metallic plate in a subsequent welding process performed by a different welding machine. The grinding assembly is moveable to displace the respective grinding surfaces from engagement with the rail when the rail cart moves along the rail and is moveable to engage the respective grinding surfaces with the vertical web of the rail when the rail cart is maintained in a stationary position relative to the rail.