Consider the angle between the tow rop and the tug's direction of travel - call this angle theta. The angle theta starts at 90 degrees and as the tug moves forward teh angle assymptoticallly approaches 0 degrees. The differential equation that controls theta is:
where 'R' = length of the tow rope and 'v' = velocity of the tug.
This solves as follows:
Raise both sides by e, and this becomes:
where 'y' is the tug's position. Attached is a plot of this angle versus the tug's position in terms of multiples of R.