How do I calculate the distance a ship will take to stop?

It appears given your other post that you have already determined something to this information you have requested.

However, I do not think, though, that your answer is going to be found in determining the jerk. You really have a drag force acting on your boat. Rather than a higher-order time component, you have an extra velocity component with your total acceleration: $$ v_f =v_i + a_{tot}t = v_i + left(a_{ship} + k v^alpha right)t $$ where $k$ is some constant and $alpha$ is a power (usually either 1 or 2).

You could begin with a law :

$$dot v = - a(1+bv^2)$$ where $a$ and $b$ are positive constants.

The integration of this gives, (see this), the formula :

$$v(t) = frac{v_0 - large frac{tan (a large sqrt{b}t)}{sqrt{b}}}{1+sqrt{b}v_0 tan(a sqrt{b}t)}$$

The ship stops at time :

$$t_{stop} = frac{tan^{-1}(sqrt{b} v_0)}{a sqrt{b}}$$

The equation for $x(t)$ is, (see this):

$$x(t) = - frac{log (sec^2(a sqrt{b}t)) - 2 log(sqrt{b} v_0 tan(a sqrt{b}t)+1)}{2ab}$$

Plugging $t_{stop}$ in this equation, we get :

$$x_{stop} = frac{log(1 + b v_o^2)}{2ab}$$

When stopping the engines, how quickly will a ship lose its speed, and how far will it go?

Newton's law tells us the change in the ship's momentum equals the drag force:

$$M frac{dv}{dt} = - F_{drag}$$

Here $M$ is the ship's mass, and $v$ is its speed. For ships with a large areal cross section $A$ under the water line and a speed $v$ such that $sqrt{v^2 A} >> nu$ with $nu$ the kinematic viscosity (momentum diffusion constant) of the water, the drag force is given by:

$$F_{drag}= frac{1}{2} C_D rho v^2 A$$

Here, $rho$ is the density of the water, and $C_D$ the drag coefficient, a dimensionless constant typically in the range 0.1 - 0.5, depending on the shape of the ship.

This is all you need. The rest is straightforward math. Substituting the equation for the drag force into Newton's law, one readily obtains

$$frac{dv}{dt}= frac{-1}{L}v^2$$

With $frac{1}{L} = frac{C_D rho A}{2M}$. The solution to this equation is $v = L/(t+t_0)$ with $t_0$ chosen such that the ratio $L/t_0$ matches the initial speed of the ship.

Clearly, although the ship will shed its speed rapidly at early times, at later times the speed loss slows down considerably. The distance travelled is the integral over $v(t)$:

$$x(t) = L ln{frac{t+t_0}{t_0}}$$

Some specific results:

If it takes a time $t_0$ and a distance $(ln 2) L = 0.693 L$ to half the ship's speed, it will take an additional time $2t_0$ and an additional distance $0.693 L$ to again half the speed. The total time to reduce the speed by 90% is $9t_0$. During that time period the ship will travel a distance of $2.30 L$

Estimation of the parameter $L$ and $t_0$ from velocity vs time data is easy: $t_0$ is the time it takes to reduce the initial speed $v_0$ to half the value, and $L_0$ is the product $v_0 t_0$.

Note that the derived results are valid up to times $t$ at which $v(t)sqrt{A} >> nu$ or $t+t_0 << L sqrt{A}/nu$.

Nice to see some ship questions around here, I'm a naval engineer!

So, you're looking for a simple, ball-park number for a question that is in reality pretty complicated. Johannes answer might give reasonable results since it's constantly updating the number; but I want to point out some assumptions made here which might affect the result's accuracy.

Background Info: First is that Johannen's $C_d$ (which in Naval Arch is usually named $C_t$ since it would correspond to the total drag coefficient) is actually described as $C_t = f_1(frac{V^2}{gL})+f_2(frac{VL}{v})$, where $f_1$ and $f_2$ represent the wave-making (residuary) resistance coefficient ($C_r$) and frictional resistance coefficient ($C_f$) accordingly, $V$ is the ship's speed, $L$ is the ship's length, $g$ is gravitational acceleration, and $v$ is water's viscosity. As you can see, it's far from constant and changes from ship to ship, strongly dependent on their length. So to have an accurate result for your computer algorithm, you would need the chart for the boat's $C_t$. But even if you had this, it would still be off (but on the conservative side) since ships fouling strongly affect $C_f$.

Answering your Question: If your speed readings updated a little quiker, you could approximate the "instantanous" $C_t$ by approximating it with a Taylor's expansion, and then setting a system of equations with Johannes third equation. However, even with a first order approximation, you would need 3 samples or 1.5 minutes to get your first reading. And this might mean that the your "accuracy" might be lagging by the same amount. So, it might be that without any prior information of the ships (and no fancy smart/learning algorithms saving/estimating information of the ships from past data), the best you could do is Johannes approach, with some few modifications so that you can get the information you are asking for:

Quick-and-Dirty Method: First (sorry for any Kosher mathematicians out there), consider that:

$$frac{partial^2 x}{partial t^2} = frac{partial }{partial x}left(frac{partial t}{partial x} right ) = frac{partial V}{partial t}left ( frac{partial x}{partial x} right ) = Vleft ( frac{partial V}{partial x} right )$$

Substituting this to Johannes third equation, and integrating using separation of variables (let's assume that Johannes $L$ is actually constant, and let's name it $alpha$) with limits of integration $(0-x_{end})$ and $(V_0-delta )$ for the $x$ and $V$ accordingly, we get:

$$x_{end} = alphalnleft(frac{V_0}{delta}right)$$

where $V_0$ would be your initial speed (in your case, your current speed), $delta$ is the speed your going to end at, and $alpha$ you assume to be a constant (but in reality you'll updated at each time step). You mentioned you want $delta$ to be zero, but as you can see this is not possible, your result would be infinity (classic example of Zeno's Paradox, as Johannes result more clearly illustrates).

You have many options to estimate $alpha$. If you get erratic results with the most basic option I'm going to present here, I recommend you look into derivative smoothing. The most basic option would be to use a numerical derivative in Johannes third equation, $$frac{V_t-V_{t-1}}{Delta t} = frac{-1}{alpha_t} V_t^2$$ Solving for $alpha$, $$alpha_t = frac{V_t^2Delta t }{V_{t-1}-V_t}$$ To make this obvious, you'll calculate at each time step $alpha_t$, and apply it on $$x_{end} = alpha_tlnleft(frac{V_t}{delta}right)$$ Now $delta$ would have to be a speed you'll reach when you're at $x_{end}$ (this result will be very ballpark, for the reasons I commented above). You mentioned zero, so a speed you'll consider small enough to be zero... perhaps 0.02 knots? But let's be real, in a river you'll have currents so you'll never really get to zero unless you're going upstream or you're facing some strong winds. So you'll have to play around with $delta$ until you get results that seem useful to you (and probably conservative as well).

As you know the wave making resistance will reduced by reducing the ship speed. but this resistance does not decrease linearly. When the ship moves at a slow speed its value is very low. And non-linearly with increasing speed wave making resistance will increase. Thus reducing the speed at high speeds faster will happen