Absolutely 
But I don’t suppose the tug provider had any of these available.
It was the same in my district where I wanted as many 80t + ASD’s or Rotors strapped on when conditions got challenging to bully my 400m box boat the way I wanted it to go , whereas my colleagues upriver would much prefer a couple of 30t Voiths or similar to nudge the overgrown daddy’s yachts like warships and STS 's around.
You have to work with what is available
In Norfolk they reach out to the smaller tug companies, this even before the big tractor tugs came out. These were not from the main ship docking tug companies.
That diagram implies far more contribution from the tug than is justified, it doesn’t take into account how the forces change over time.
I don’t know how to solve the problem but simple vector analysis is not sufficient, it would require differential equations.
KC,
I agree that it is very difficult to accurately quantify the vector and quantifying it is a moot point. The magnitude of the vector is irrelevant.
There are some members here who believe that the tug did not contribute to the sternway and that is fine. I firmly believe that it did and the reason that I raised it is, as professionals, we are seeking causation. This is but one factor.
In my patch in the UK you have to be certified as a Ship Assist Tug so the smaller barge towing tugs could not be utilised.
For as often as they would be required I don’t suppose it was financially prudent to maintain the certification and only be used once in a blue moon
Edit to add
And have a current certified Bollard Pull
I agree. As I told the other fellow, I was theorizing what might have went wrong, not what should have been done for it to go right.
Read that like 5 times - no clue what you are telling me - sorry
The drawing with vector sollutions as provided by @By-C is a bit exaggereted but it perfectly demonstrates the tug influence . Since we do not know exactly the CPP setting and what speed/thrust it represented and obviously we do not know the tug power setting the drawiing has only illustrative value.
And whom would You call here to solve this equations??? Only alias @Earl_Boebert1 come to my mind and for this equations one needs some values too.
added: how about tug’s angle of attack?? these were not constant all the time but were variable as well as forces/thrusts applied. T|he object trajectory was not a straight line either . It was a some kind of curvature the shape of which can only be extracted by carefull plotting . As far as I am concerned it is too much work .Let NTSB Gurus take care of that .
Read yours 6 times and figured it would be proportional answer . 
late to the party - love all the vector diagrams - but with the little experience I have - I can’t see how that tug - without a line up - can add any appreciable sternway at all. I don’t care what he was doing to stay at a 90 - without a line up - all he could add was rotational force.
I was serious - wasn’t trying to be snarky - but never mind - doesn’t matter.
Well… we have several pilots here .Let them elaborate 
Ithink I am not up to the task. My extensive experience is only standing on the bridge
Added . BTW why do U think we put woodden dunnage or sometimes rubber dunnage between heavy pcs of cargo and toptanks/ H.Covers surface when cgo is to be deposited. ???. The pproper answer is to increase the friction.
I think analyzing the various forces is probably not worthwhile. More likely the focus will be on BRM wrt trapping the error with the CPP in a timely manner.
Sal’s commentary very suggests that he knows exactly who the relevant pilot is incidentally.
You may be right. Dr.Sal surely has sources not available to average Joe like me . Observation: You do not sound (sorry!!!) read like a Londoner . This makes two of us here
It’s starting sound like some people think those damned gringo pilots had some magic device that jammed the CPP in full astern.
Yea - at least from my perspective the question isn’t what happened - pretty sure we know the CPP was full astern is what happened.
The question now is when, as in when did it happen? When did the pilot/crew notice?
and why - why did it happen ?
My guess is the investigation will find some stuff people could have done better, but the overwhelming cause will be the mechanical failure of the CPP at a really bad time.
The 3 WHYs is a truncated method in investigation.
The full version contains 5 WHYs
THANKS!!! I need to hug You Aus. How can we arrange that ???
My guess would be the order ahead was given about the time the tug was let go.
Anyone know if the prop rotation was LH or RH? Given the way the bow swung LH makes more sense.