Optimal amplitude modulations for a radar signal are derived and then used to calculate the efficiencies of various sub-optimal modulations. The choice of modulation is constrained by the total energy transmitted and the peak power (amplitude) of the transmitted signal. The peak power constraint is handled by the use of Pontryagin's Maximum Principle, an extension of the calculus of variations recently developed in the U.S.S.R. that is enjoying wide application in optimal control theory. The criterion of optimality is based on the error variances of estimates of the range motion parameters of a reflecting body, where the errors are caused by additive, white, zero mean, Gaussian noise. Explicit results are provided for bodies with constant velocity and bodies with constant acceleration. The analysis covers: 1) incoherent processing of a sequence of many range measurements; 2) coherent processing assuming the RF phase is known, 3) certain aspects of coherent processing assuming the RF phase is unknown. The optimal modulations turn out to be of the "on-off" type, requiring either no transmission or transmission at the maximum allowable power level.