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I am having trouble writing down the coordinate systems for a problem. In particular, I'm not sure how to figure out the translational acceleration of the moving coordinate system relative to the fixed. The formula I'm using is: $\vec{F_{r}}=m\vec{a_{r}}=\vec{F_{f}}-m\ddot{R_{f}}-m \vec{w}\times \vec{r}-m \vec{w}\times (\vec{w} \times \vec{r})-2m\vec{w} \times \vec{v_r}$

And I am having trouble identifying $\ddot{R_f}$.

The problem is: An automobile drag racer drives a car with acceleration a and instantaneous velocity v. The tires (of radius $r_0$) are not slipping. Find which point on the tire has the greatest acceleration relative to the ground. What is this acceleration?

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Assume that the origin of frame $r$ is moving with acceleration $\newcommand{\a}{\mathbf{a}}\a$, with respect to a fixed frame $f$, but that the coordinate axes of $r$ are aligned with $f$'s. Then if an object has an apparent acceleration $\a_r$ in the frame $r$, then it's actual acceleration $\a_f$ in the fixed frame is $\a_r + \a$.

Now when your allow the coordinate axes of $r$ to move, then you pick up the other forces. But hopefully the first paragraph answer your question.

To solve the motivating question. I would pick the frame $r$ to be in the frame moving with the car, but with axes fixed. Then $\a_r$ goes around in a circle, and $\a_f$ is just $\a+\a_r$, where $\a$ presumably points in the direction of motion of the car. So the maximum of $\a+\a_r$ is when they point in the same direction. I will let you do the rest.

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