Considering a 2D motion in a plane and the equation of the trajectory of a point $y=f(x)$, I don't understand how exactly $\frac{\mathrm{d}y}{\mathrm{d}x}$ can be used.

In particular if I'm looking for the tangential acceleration, I need the tangential unit vector $u_T$.


$$\lvert\vec{v}_y \rvert=\frac{\mathrm{d}f(x(t))}{\mathrm{d}t}=\frac{\mathrm{d}f(x)}{\mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}t}=\frac{\mathrm{d}y}{\mathrm{d}x} \lvert \vec{v}_x \rvert$$

Can I conclude that


If so, can I do something similar for trajectories (like circles) in the form $f(x,y)=0$?

Furthermore how can I get the normal unit vector $u_N$ (for istance to find the normal acceleration) from the $u_T$ previously found?

I know that $u_N=\frac{\mathrm{d}u_T/\mathrm{d}t}{\lvert\mathrm{d}u_T/\mathrm{d}t\rvert}$ but the derivative is with respect to time here.

Am I missing something?


At every point the tangential direction is the unit vector of the velocity vector. If you have the velocity components $\boldsymbol{v} = (\dot{x}, \dot{y})$ at every instant, the you decompose this into a magnitude (speed $v$) and direction $\hat{\boldsymbol{e}}$

$$ \begin{align} v & = \sqrt{\dot{x}^2+\dot{y}^2} \\ \hat{\boldsymbol{e}} & = \begin{pmatrix} \frac{\dot{x}}{v} & \frac{\dot{y}}{v} \end{pmatrix} \\ \boldsymbol{v} & = v\; \hat{\boldsymbol{e}} \end{align} $$

The normal direction is simply

$$ \hat{\boldsymbol{n}} = \begin{pmatrix} -\frac{\dot{y}}{v} & \frac{\dot{x}}{v} \end{pmatrix} $$


The easy way of doing this is to parametrice the trajectory. We have the cartesian definition, so let $ \textbf{r} : \mathbb{R} \rightarrow {\mathbb{R}}^{2} $ be: $$ \textbf{r}(t)=(t,f(t)) \quad\quad t\in (-\infty,\infty) $$

So we got the vector position as function of "time". Then, the velocity and acceleration vectors are defined by: $$ \textbf{v}(t)=\frac{d}{dt}\textbf{r}(t) \quad \quad \textbf{a}(t)=\frac{d}{dt}\textbf{v}(t) $$

Now, you can get the tangential acceleration with the projection over the velocity, that is always tangential to the trajectory: $$ {a}_{t}=\textbf{a} \cdot \frac{\textbf{v}}{v} $$ Where $v$ is the length of the velocity vector.


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