Since acceleration is invariant under constant-velocity Galilean transformations, a system moving at constant velocity cannot, in Newtonian physics, exert new forces it doesn't already exert when it is at rest. But a bar magnet moving at nonzero constant velocity exerts a force on electric charges that it doesn't exert when it is at rest (Faraday's Law), and a charge moving at nonzero constant velocity exerts a torque on the needle of a magnetic compass that it doesn't exert when it is at rest (Biot-Savart Law). Thus basic electromagnetic experiments which are feasible in undergraduate or secondary-school physics labs illustrate the need to replace the Galilean transformations. That seems pedagogically much more compelling than the standard practice of merely discussing experiments which use extremely high-precision equipment such as Michelson interferometers. What should replace the Galilean transformations? A key qualification obviously is compatibility with the electromagnetic Laws. Those Laws can be presented as wave equations with source terms, and wherever the source terms are zero, the free waves travel exclusively at the fixed constant speed c. Thus to be compatible with the electromagnetic Laws, the replacements of the Galilean transformations must preserve the speed c, but they of course must in addition become Galilean when the ratio of the untransformed speed to c goes to zero.