Bullet Train: Coriolis Force

The Japanese Shinkansen train (bullet train) zips from Tokyo to Osaka (both at approximately \(35^{\circ}N\)) at a speed of \(185\text{ }km\text{ }h^{-1}\). In the design of the train and tracks, do you think that engineers had to worry about Earth’s rotation?

SourceCushman-Roisin and Beckers: Physical problems 2-2.

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If the train is at \(35^{\circ}N\), then the Coriolis parameter is:

$$\begin{align*}f &= 2\Omega\sin(35^{\circ}) \\ &\approx 2\left( 7.2921 \times 10^{-5} \right)\sin(35^{\circ}) \\ &\approx 8.37 \times 10^{-6}\text{ }s^{-1}\end{align*}$$

This exhibits a transverse force proportional to the speed of the train:

$$fU = 185\text{ }km\text{ }h^{-1} \times f \approx 4.3 \times 10^{-3}\text{ }m\text{ }s^{-1}$$

Combined with the downward gravitational acceleration of \(9.81\text{ }m\text{ }s^{-2}\), this exhibits a tilt of:

$$ \theta \approx \text{arctan}\left( \frac{4.3 \times 10^{-3}}{9.81} \right)\approx 4.3 \times 10^{-4}$$

This is insignificant, and thus not a concern for an engineer.