We sit on a rotational ellipsoid spinning around its axis roughly every 23 h 56 min. What is the speed at which we travel on the surface at a given latitude?

Let's center our planet on a three-dimensional coordinate system.
The axis of rotation is the *z*-axis, *x*- and *y*-axis follow from the right hand rule.
If we make a cut at *y* = 0, we get an ellipse with a semi-major axis *a* = 6378 km and a semi-minor axis *b* = 6357 km.

Given a latitude \(\phi\), we can set up a line through the origin with a slope \(m = \tan (\phi \cdot \frac{180}{\pi}).\)
$$
z = m x
$$
This line intersects the ellipse centered at \(O = (0, 0)\) in point \(P\).
$$ \frac{x^2}{a^2} + \frac{z^2}{b^2} = 1 .$$
By substituting the linear equation into the equation of the ellipse, we can compute the *x*-component of point \(P\), which is the radius \(r\) of the circular arc on which somebody sitting on the surface at latitude \(\phi\) is spinning around the axis.
$$
x = \sqrt{\frac{a^2 b^2}{b^2 + m a^2}} = r
$$
Assuming constant circular speed, we get the velocity by computing the circumference and dividing it by the time for a full rotation.
Take, for example, 45° latitude, somewhere north of Bordeaux, France.
$$ v = \frac{2 \pi r}{\Delta t} \approx \frac{2 \pi \cdot 4502 \text{km}}{23.93 h} \approx 1180 \frac{\text{km}}{\text{h}} $$

Hold on tight! It's a rough ride.