Obtain Radial and cross radial velocity and acceleration

Question

Request: My dynamics book does not contain the vector method to obtain the radial and cross-radial velocity and acceleration. Does anybody give me the proof, please? I have searched the internet but found nothing useful.

Thank you for your help.

Answer 1

For dimension two, the calculation is as simple as to write the velocity in polar coordinates. In dimension three, playing with the velocity ($\dot{\vec r}$) and the position vector, we can write

$\vec v_r=\left(\dfrac{\vec r}{r}·\dot{\vec r}\right)\dfrac{\vec r}{r}=\dfrac{1}{2r^2}\dfrac{d(r^2)}{dt}\vec r$

$\vec v_{cr}=\dot{\vec r}-\vec v_r$

$\vec a_r=\left(\dfrac{\vec r}{r}·\ddot{\vec r}\right)\dfrac{\vec r}{r}=\dfrac{1}{r^2}\left(\dfrac{d(\vec r·\dot{\vec r})}{dt}-\dot r^2\right)\vec r$

$\vec a_{cr}=\ddot{\vec r}-\vec a_r$