Note that \(\hat{r}\) is a unit vector that points in the direction from the origin to the point in question, and \(\hat{\theta}\) is the unit vector that points in the perpendicular direction, going counter-clockwise.
2.2.1 Write \(\hat{r}\) and \(\hat{\theta}\) in terms of \(\theta, \hat{x}, \hat{y}\).
2.2.2 Take the time derivative of the equations obtained in 2.2.1. Note that \(\hat{x}\) and \(\hat{y}\) are constant with respect to time. Simplify the equations to remove x and y.
2.2.3 Hence, differentiate \(\vec{r}\) by application of the product rule. Check if the answer makes sense.
2.2.4 Hence, differentiate \(\dot{\vec{r}}\) by application of the product rule to obtain \(\ddot{\vec{r}} = (\ddot{r}-r\dot{\theta}^2 )\hat{r} + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{\theta}\).
Allow us to analyse the results obtained. The first term is simply the component of the acceleration in the radial direction, and likewise for the third term in the tangential direction. The second term is familiar to us -- the centripetal acceleration. The last term isn't as obvious -- this is what gives rise to the Coriolis force.