A body that can be regarded as a point mass is sliding down a smooth hill under the influence of gravity. The hill is the graph of a differentiable function $y = f (x)$.
a) Find the horizontal and vertical components of the acceleration vector that the body has at the point $(x_0, y_0)$.
b) For the case $f(x) = x^2$when the body slides from a great height, find the point of the parabola $y = x^2$ at which the horizontal component of the acceleration is maximal.
This is a problem from Zorich Mathematical Analysis I. I noticed that there has been one question asked about this, but I am not sure if it answered what it meant to be answered from the author. Here is the link: Differentiable Functions / Zorich
In the answer from the link, the answer which is only for the first part says that:
The mass is always on the graph. The acceleration is directed down the slope (downhill down the tangent), and basic physics tells you, that the magnitude of the acceleration is proportional to the sine of the slope angle.\A normalized vector along the tangent is $(\cos\alpha,\sin\alpha)$, when expressed with slope angle $\alpha$. Acceleration is thus$$\vec{a}=-(\cos\alpha\sin\alpha,\sin^2\alpha)$$
Take into account $\tan\alpha=f'(x)$ and you get$$\vec{a}=-\frac{(f'(x),f'(x)^2)}{1+f'(x)^2}$$
My question is as follows:
Is this correct? If it is correct please help me justify it. If it is not, why is that? And how I should approach this and the ensuing one?