mechanics is hard...need help!!

[teru]

hello physics forums,
this is lengthy but pretty much challenging..really appreciate any help ^o^

A tugboat is towing a barge with a flexible cable. After towing steadily at constant speed for sometime, the tug propeller thrust T(t) is decreased from 100kN at a steady rate (dT/dt = constant).

given: mass of tug, m1 = 10,000 kg
mass of barge, m2 = 100,000 kg
tug drag characteristic, D1 = 2000.(v1)^2 N
barge drag charateristic, D2 = 6700.(v2)^2 N
cable stiffness 800 kN/m
vb = 0.4904 m/s

1) Determine the maximum magnitude of the 'steady rate' that ensures the tow cable is always in tension during the deceleration from the constant speed to a barge velocity, vb m/s (ignore the cable mass and its sag).

2) Can the non-linear differential equations be solved numerically. how will v1 and v2 behave with respect to time, t?

[HallsofIvy]

1) I'm not sure how "cable stiffness" affects this but:
Let F be the tension in the cable. There are three forces on the tugboat: the thrust from its propellor, T, which is positive, the drag, which is negative, and the pull from the barge, F, which is negative: 10000\frac{dv_1}{dt}= T- 2000v_1^2-F.
There are two forces on the barge, the pull from the tugboat, F, which is positive, and the drag, 6700 v22, which is negative: 100000\frac{dv_2}{dt}= F- 6700v_2^2.

Question 1 asks, what is the maximum of dT/dt so that F never becomes 0.