In any context where the speed of travel is virtually (or, in this case, exactly) a constant, people automatically begin to express distances in time units. [Q: ``How far is is from New York to Boston?" A: ``Oh, about three hours."] This is equivalent to defining the speed of travel to be a dimensionless constant of magnitude 1. Relativistic Physics is no different. Anyone who has to discuss relativistic phenomena at any length will usually slip into ``natural units" where
is just ``the velocity" measured in natural units
and the calculations become much simpler.
But you have to convert all your other units accordingly,
and this can be interesting. It does take a little getting
used to, but the exercise is illuminating.
I will try to do it both ways (with c as a constant in velocity units
and with c = 1 and unitless) wherever possible, just to give you
a feeling for how this goes.