February 10, 2022

Notation used :

Time derivatives are sometimes denoted by dot (therefore $\frac{d^2x}{dt^2} = \ddot{x}$),

$\clubsuit$

Change in a Vector: Consider a general vector $\vec{A}$, since it has both magnitude and direction, we can create a change in both of them, that is there can be cases when the length of the vector may not change, but its direction would.

If we add a vector $\Delta\vec{{A}}\perp$, perpendicular to $\vec{A}$, then $\vert\Delta\vec{{A}}\perp\vert$ $\approx$ $\vert\vec{A}\vert{\Delta{\theta}}$ (via small angle approximation), where ${\Delta{\theta}}$ is the angle between $\vec{A}'$ and $\vec{A}$, where $\vec{A}'$ = $\vec{A}$ + $\Delta\vec{{A}}\perp$ $\rightarrow$ $\vert\vec{A}'\vert$ = $\vert\vec{A}\vert$[1+${(\Delta\theta)}^2$]. In $\lim{{\Delta\theta\to 0}} \vert\vec{A}'\vert$ = $\vert\vec{A}\vert$, that is the magnitude of the vector remains unaltered because $\Delta\vec{{A}}_\perp$ is too small to change the magnitude. It just pushes the vector $\vec{A}$ so to speak.

             Figure 1.1

Now taking the general case, where we add a vector $\Delta\vec{{A}}$ to $\vec{A}$ at an angle $\Delta\theta$ WRT $\vec{A}$. Therefore

$$ \Delta\vec{{A}}=\Delta\vec{{A}}\perp+\Delta\vec{{A}}\parallel $$

              Figure 1.2

Again since in $\lim$$\Delta\theta$$\rightarrow$0, $\vert\Delta\vec{{A}}\perp\vert$ $\approx$ $\vert\Delta\vec{{A}}\vert{\Delta{\theta}}$ (since this condition is same as discussed above) and $\vert\Delta\vec{{A}}\parallel\vert$ = $\vert\Delta\vec{{A}}\vert$, Differentiating both vectors WRT time we get

$i)\quad$$\vert\frac{d\vec{A}\perp}{dt}\vert$ = $A\frac{d\theta} {dt}$ and $ii$) $\vert\frac{d\vec{A}_\parallel}{dt}\vert$ = $\vert\frac{{d{A}}} {dt}\vert$

In other words, a $\perp$ change to a vector applied in a very small duration can only change its direction but not magnitude. The key thing here is small duration, because otherwise if we continued to apply the change in the initial direction, the vector’s magnitude will increase.

One can use the above-given results to derive equations of velocity and acceleration in polar coordinates.

Kinematics of a point: The natural method

Suppose we know the path of the particle beforehand, i.e. path is described by a function l, which is itself a function of time, therefore