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The current is maximum through those segments of a circuit that offer the least resistance. But how do electrons know beforehand that which path will resist their drift the least?

Ohm's law is represented by the equation $I=\frac{V}{R}$. How would the current change if the amount of resistance decreased and the voltage stayed the same?

A submarine of mass m is travelling at max power and has engines that deliver a force F at max power to the submarine. The water exerts a resistance force proportional to the square of the submarine's speed v.
The submarine increases its speed from $v_1$ to $v_2$, show that the distance travelled in this period is $\frac{m}{2k}\ln \frac{F-k{v}_1^2}{F-k{v}_2^2}$ where k is a constant.
I've found $\frac{dv}{dt}=\frac{1}{m}(F-k{v}^2)$ using $\sum F=ma$ but I am struggling to progress further using integration or the fact that $\frac{dv}{dt}=v\frac{dv}{dx}=\frac{d}{dx}(\frac{1}{2}{v}^2)=\frac{d^{2}x}{d{t}^2}$ which is how I've been taught to solve resisted motion questions.

What is the electric current produced when a voltage of 15V is applied to a circuit with a resistance of $6\mathrm{\Omega }$ ?

The equations are as follows:
${\rho }_a(s)=s^2{\int }_0^{\mathrm{\infty }}T(\lambda )J_1(\lambda s)ds(4)$
Where ${\rho }_a$ is the apparent resistivity and $T(\lambda )$ is a resistivity transform function. Using a Hankel Transform, we get:
$T(\lambda )={\int }_0^{\mathrm{\infty }}{\rho }_a(s)J_1(\lambda s)\left[\frac{1}{s}\right]ds(7)$
New variables are then defined by
$x=ln(s)andy=ln\left(\frac{1}{\lambda }\right)(8)$
Substituting (8) in (7), we get
$T(y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\rho }_a(x)J_1\left(\frac{1}{e^{y-x}}\right)dx(9)$
Please note that any missing equations were for another resistivity equation.