A potential difference of 14 V is found to produce a current of 0.45 A in a 3.8 m length of wire with a uniform radius of 0.36 cm. Find the following values for the wire: (a) the resistance Omega (b) the resistivity Omega * m

How do you calculate voltage drop in a series circuit?

Determine the resistance of metal tube in term of the external diameter D, the internal diameter d , the length l and the resistivity. Calculate the resistance of a copper tube 0.5cm thick and 2m long. The external diameter is 10cm

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 electric potential across a resistor if a resistance of 20ohms draws a current of 6 amperes?

This is kind of a stupid question and I am taking some risk of getting some down-votes here, but, I can't resist posting it. Suppose $(u_1,u_2)$ is an orthonormal basis for $R^2$, and let x be an arbitrary vector in $R^2$, then we can decompose x by projecting on $u_1,u_2$, i.e.
$\begin{array}{r}(u_1^{T}x)u_1\\ (u_2^{T}x)u_2\end{array}$
Certainly we have
$\begin{array}{r}x=(u_1^{T}x)u_1+(u_2^{T}x)u_2\end{array}$
Looks like a simple problem, but how can I prove this?