A particle is moving within a curved trajectory defined in the plane $(xoy)$, with the polar coordinates : $r(\theta) = r_0(1-cos\theta)$ with $t=\omega t$ with $r_0$ and $\omega$ positive constants.

First question is to give the positional vector $\vec{OM(t)}$ and velocity $\vec V(t)$ and acceleration $\vec a(t)$

I feel like what i did is wrong : what i did is i simply put $\omega t$ instead of $\theta$ in the equation $r(\theta)$ and we have $\vec {OM(t)} = r(t).\vec U_r$ ($\vec U_r$ unit vector).

and then i continue deriving to get $\vec V(t)$ and $\vec a(t)$., but is there a way to transform $r(\theta)$ to cartesian equations $x(t),y(t)$? i know that $x(t) = r(t).cos \theta$, but does it work in this case?

Second question is to give the algebraic value of $a_t$, now i worked with both unit vectors $\vec U_r,\vec U_\theta$, and i know that $a_t$ is the derivative of the magnitude of velocity, so all i have to do is derive $v(t) = \sqrt{V_r^2(t)+V_\theta^2(t)}$ and get it as a function of time? this looks wrong to me.

I'm having troubles with polar coordinates and i really hope to get some references to some good exercises/books to get my self used to working with them.

A particle is moving within a curved trajectory defined in the plane $(xoy)$, with the polar coordinates : $r(\theta) = r_0(1-cos\theta)$ with $\theta=\omega t$ with $r_0$ and $\omega$ positive constants.

First question is to give the positional vector $\vec{OM(t)}$ and velocity $\vec V(t)$ and acceleration $\vec a(t)$

I feel like what i did is wrong : what i did is i simply put $\omega t$ instead of $\theta$ in the equation $r(\theta)$ and we have $\vec {OM(t)} = r(t).\vec U_r$ ($\vec U_r$ unit vector).

and then i continue deriving to get $\vec V(t)$ and $\vec a(t)$., but is there a way to transform $r(\theta)$ to cartesian equations $x(t),y(t)$? i know that $x(t) = r(t).cos \theta$, but does it work in this case?

Second question is to give the algebraic value of $a_t$, now i worked with both unit vectors $\vec U_r,\vec U_\theta$, and i know that $a_t$ is the derivative of the magnitude of velocity, so all i have to do is derive $v(t) = \sqrt{V_r^2(t)+V_\theta^2(t)}$ and get it as a function of time? this looks wrong to me.

I'm having troubles with polar coordinates and i really hope to get some references to some good exercises/books to get my self used to working with them.