Is this an acceptable use of $a \,dx = v \,dv$?
From the chain rule we have
$$a\, dx = \frac{dv}{dt}\, dx = \frac{dv}{dx} \frac{dx}{dt} \,dx = v \frac{dv}{dx} dx = v \,dv$$
This was introduced in an early chapter of my physics textbook and I have been getting a lot of mileage out of it. For example, in part (b) of this problem
A small object is placed at the top of an incline that is essentially frictionless. The object slides down the incline onto a rough horizontal surface, where it stops in 5.0 s after traveling 60 m. 563
(a) What is the speed of the object at the bottom of the incline and its acceleration along the horizontal surface? (b) What is the height of the incline?
you are supposed to use the work-energy theorem, like so:
Let $x$ denote the hypotenuse of the incline and $\theta$ its slope. Then $mg \sin\theta$ is the component of $F_g$ parallel to the surface.
$$\begin{align}W_\mathrm{net} &= \Delta K \ mg \,x \sin\theta& = \frac{1}{2}m(v_1^2 -0) \end{align}$$
Given $v_1 = 24$ from part (a), we can solve for the height $x \sin\theta = 29.36$ m.
Instead, I noticed that $a\,dx = v\,dv$ implies $\bar a \Delta x = \bar v \Delta v$. The parallel component of the acceleration vector is $g \sin \theta$, $\Delta x = x$, $\bar v = \frac{1}{2}(v_1 - 0) = \frac{v_1}{2}$, and $\Delta v = v_1$. Then we have
$$\begin{align} \bar a \Delta x &= \bar v \Delta v \ g \,x \sin\theta & = \frac{v_1^2}{2} \end{align}$$
which gets you the same thing.
* Are there situations in which an approach like this wouldn't work? (I realize that I assumed constant acceleration when calculating $\bar v$, but anything beyond that?)
* Is it correct (from a physics standpoint) to say that $a\,dx = v\,dv$ implies $\bar a \Delta x = \bar v \Delta v$? I would prove this mathematically by integrating on both sides, noting that the average value of a function $f(x)$ on an interval $(x_0, x_1)$ is $$\bar f(x) = \frac{1}{x_1 - x_0}\int_{x_0}^{x_1} f(x) dx$$
* Does my work amount to a derivation of the work-energy theorem, or a lucky circumvention of it?
* Silly question: If a student did this on a test, would you give them credit? 1002