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Figure 4-1. Free-body diagram of ship
Notice here that the buoyancy force is exactly equal in magnitude to the ship's weight
and opposite in direction; thus, these forces cancel each other out and there will be no
motion in the y-direction. This must be the case if the ship is to stay afloat. This obser‐
vation effectively reduces the problem to a one-dimensional problem with motion in
the x-direction, only where the forces acting in the x-direction are the propeller thrust
and resistance.
Now you can write the equation (for motion in the x-direction) using Newton's second
law, as follows:
ΣF = m a
T - R = m a
T - (C v) = m a
Where a is the acceleration in the x-direction, and v is the speed in the x-direction.
The next step is to integrate this equation of motion in order to derive a formula for the
speed of the ship as a function of time. To do this, you must make the substitution a =
dv / dt , rearrange, integrate, and then solve for speed as follows:
T - (C v) = m (dv/dt)
dt = (m / (T-Cv)) dv
(0 to t) dt = ∫ (v1 to v2) (m / (T-Cv)) dv
t - 0 = -(m/C) ln(T-Cv) | (v1 to v2)
t = -(m/C) ln(T-Cv 2 ) + (m/C) ln(T-Cv 1 )
t = (m/C) [ln(T-Cv 1 ) - ln(T-Cv 2 )]
(C/m) t = ln [(T-Cv 1 ) / (T-Cv 2 )]
e (C/m) t = e ln [(T-Cv1) / (T-Cv2)]
e (C/m) t = (T-Cv 1 ) / (T-Cv 2 )
(T-Cv 2 ) = (T-Cv 1 ) e -(C/m)t
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