Vehicle moving along a path tangent to the Earth surface

[Like Tony Stark]

Homework Statement

A vehicle moves along a path perfectly tangent to the earth surface in the Equator. The path doesn't have any curvature in the vertical plane. What velocity should have the vehicle with respect to the path so that its vertical acceleration component is zero? (consider that the Earth centre is not accelerated). Then, how would your previous answer change if the vehicle moves along a path that has the same curvature that the Earth?

Relevant Equations

$\vec a=\vec a_B + \vec{\dot \omega} \times \vec r + \vec \omega \times (\vec \omega \times \vec r) + 2. (\vec \omega \times \vec v_{rel}) + \vec a_{rel}$

Here I have some problems. I get confused when it says"with respect to the path", is it different from "with respect to the earth"? Because the path is on the Earth. Then, the vehicle is not accelerated in the vertical direction because it moves along the path, is it?

 

[TSny]

The path is attached to the Earth and rotates with the earth. You need to determine the magnitude and direction of the velocity of the vehicle relative to the Earth so that, relative to the non-rotating inertial frame, the car has zero acceleration in the vertical direction (zero acceleration in the y-direction at the instant shown in the figure below).