We can get into the numerical details of determining orbits preand postNewton, as well as Gauss, but there are never any units of speed used at first. What you do at the observatory is note the time and apparent position of the object using some coordinates like azimuth and altitude above the horizon; the appropriate units are degrees.
Then you can correct for various effects and express the observation in terms of an appropriate celestial coordinate system with respect to the centre of the Earth or of the Sun, so, basically, more angles; you can see why trigonometry comes in.
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