Example: Deflection of Cannon Balls

People have been known to fire cannon balls in many direction and in many locations near the earth. Let us consider the effect of the Coriolis force on the ballistic trajectories of these cannon balls. Assume the cannon ball is fired at an angle \bgroup\color{black}$ \theta$\egroup from the vertical and at and angle \bgroup\color{black}$ \phi$\egroup from the easterly direction in the plane tangent to the earth. These are the usual angles of a spherical coordinate system.

\bgroup\color{black}$\displaystyle \vec{a}=-2\vec{\omega}\times \vec{v}=-2\omega...
...{z}+\sin\theta\cos\phi\hat{e}_{east}+\sin\theta\sin\phi\hat{e}_{north}) $\egroup


\bgroup\color{black}$\displaystyle \vec{a}=2\omega v(-\sin\lambda\sin\theta\cos\...
...os\lambda\cos\theta\hat{e}_{east}+\cos\lambda\sin\theta\cos\phi\hat{z}) $\egroup

This can be summarized in three effects.
  1. cannon balls will curve to the right in the northern hemisphere and to the left in the southern hemisphere. (first two terms)
  2. There is a force to the west on the way up and a force to the east on the way down, like the falling object in the example above. (third term)
  3. cannon balls curve up when fired to the east and down when fired to the west. (fourth term)
The magnitude of these effects can be large and were accommodated in the sighting of naval cannon by using tables.

Jim Branson 2012-10-21