We consider one-dimensional thermoelastic contact problem on vertical indentation of a rigid thermally insulated half-plane moving horizontally with constant speed over an elastic coating (strip), while bottom side of the latter is bonded to a rigid foundation. Thermal flux generated by friction is directed to the strip. Temperature, displacement and stress distributions along the depth of the coating are derived in the form of infinite series over eigenfunctions. It is shown that the thermoelastodynamic instability of the obtained solutions is present in all time range and at any velocities of the half-plane sliding over the surface of the coating.