Why is "work done/time taken" correct over "force × velocity" as a definition of power? (A level) I came across a multiple choice A level past paper question (CIE June 04 paper 1 Q15) asking to choose which of the following defined power:

15 What is the expression used to define power?
(A) $\frac{\text{energy input}}{\text{energy output}}$
(B) $\text{energy × time taken}$
(C) $\text{force × velocity}$
(D) $\frac{\text{work done}}{\text{time taken}}$

The answer given was (D), but I am perplexed as to why (C) was not considered correct.
 A: D is the correct definition. C is valid only for mechanics. For instance, you cannot find the power of electrical appliances using $P=Fv$. Though the definition in option D is always correct. More precisely, D is the definition of Power while C is a derivation.
A: I am also perplexed. Both are correct, but $P=\vec F \cdot \vec v$ is better, in my opinion, because it gives you the instantaneous power. In contrast $P=\Delta W/\Delta t$ gives you only the average power.
A: Both seem to be correct, as another answer also points out. Meaning, both formulas can be used to calculate power.
But the fundamental definition of power is the latter, namely energy per time.
In this general definition, power can be calculated for mechanical systems, thermodynamic systems, electric system etc. The former version includes force and might thus be harder to use in, say, thermodynamic problems.
A: The strong definition of power is input times output, or, flow times effort. This is valid in many physical domains, electrical, mechanical, etc. The reason that flow (velocity, current, or any other rate of changes) is selected as the output is because it gives us real values (pure dissipation). Position and acceleration contributes to complex values (phase shift only and no dissipation, meaning no work is actually done).
It is definitely a bad question.
A: It's because the question asks for the 'definition'
Definition of power
so D is the answer, even though C is a valid formula.
A: Power is "defined" as work done per unit time , so option (d) is correct.
Now coming to the third option. Look the instaneous power can be given as
$$P=\frac{dw}{dt}=\frac{d(\vec F\cdot \vec s)}{dt}= \vec F\cdot \vec {v}+ \vec {s}\cdot \frac{\vec {dF}}{dt}$$
So power = force × velocity is incomplete . May be that's why they made the third option wrong.
A: Within the frame of reference of A-level physics, I think they want you to reject "force × velocity" because they're expecting you to associate the term "velocity" only with a vector quantity, not with speed.  This is a pedantic point of terminology that they certainly emphasised in my day.
They've used the division sign "÷" in answer D, so in that context I reckon there's no doubt that, by the multiplication symbol "×" in answer C, they mean some sort of elementary multiplication.  Either the product of two scalars, or of one vector and one scalar.  They want you to notice that you can't multiply (in that sense) two vectors like force and velocity, and so reject that answer.  But even if you interpret it to mean the cross product instead, that makes the answer even wronger.
Within the domain of A-level physics as taught, I think "force × speed" would have been an acceptable answer (with answer D correspondingly replaced by something wrong).
A: Velocity and force are both vectors (in this case, a number each for forward, left and up). The standard ways to multiply vectors are $F \cdot v$, which yields a scalar, the cross product, $F \times v$, which yields a vector and an outer product, $F \otimes v$, which yields a matrix.
This site on A levels covers vectors and there's no mention of how to multiply. Since they didn't say how to do it, don't. FYI, dividing vectors is meaningless and illegal forever. This site makes a baffling and unforgivable abuse of notation describing power in terms of velocity and force, but it's clearly a derivation.
If you're going to mention the $F \cdot v$ version of power, the germ of vectors must be included: it's the difference between the accelerator and the brake (ignoring the fact that the friction between the wheels and the road move the car).
