  The energy-time uncertainty relation puts a fundamental limit on the
precision of radars and lidars for the estimation of range and velocity. The
precision in the estimation of the range (through the time of arrival) and the
velocity (through Doppler frequency shifts) of a target are inversely related
to each other, and dictated by the bandwidth of the signal. Here we use the
theoretical toolbox of multi-parameter quantum metrology to determine the
ultimate precision of the simultaneous estimation of range and velocity. We
consider the case of a single target as well as a pair of closely separated
targets. In the latter case, we focus on the relative position and velocity. We
show that the trade-off between the estimation precision of position and
velocity is relaxed for entangled probe states, and is completely lifted in the
limit of infinite entanglement. In the regime where the two targets are close
to each other, the relative position and velocity can be estimated nearly
optimally and jointly, even without entanglement, using the measurements
determined by the symmetric logarithmic derivatives.
