Mechanics: Coordinate systems and vector's

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Homework Statement

An ant walks from the inside to the outside of a rotating turntable. Write down it's velocity vector.

Use polar the cartesian coordinates.

Homework Equations

I have already derived the velocity vector in polar coordinates which is:

\hat{v} = \dot{r}\hat{r} + r\dot{\vartheta}\hat{\vartheta}

The Attempt at a Solution

The table is rotating at a velocity \hat{v} whilst the ant we assume just walks in a straight line along direction\hat{j} in it's reference frame there is nothing odd, it is walking in a straight line. However in the observers reference frame is is moving in a circle due to the motion of the turntable.. So do I take it's cartesian velocity vector \dot{r}\hat{r} and simply add it with the velocity of the turntable \dot{r}\hat{r} + r\dot{\vartheta}\hat{\vartheta} ??

I'm not quite sure how this works.

 

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Ah I've done it in polar coordinates. I was half right about the addition, but I was unsure about the angular velcity of the turntable. Now doing it in cartesian.

 

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Unsure about cartesian coordinates... I've considered the velocity of the ant in x and y directions Vx = ucos theta
Vy = usin theta

Then I considered a point on the turntable which has an angular velocity rw...

I don't know what to do now...