when a long-range cannonball is fired towards the equator from a northern (or southern) latitude, it lands west of its "intended" longitude. Why?

its east-west velocity is less than the velocity of points on the equator.

its angular velocity is the same at any latitude, but the higher the latitude, the less its linear speed, because the radius of its circle is less.

in 2-D, consider a ball on a rotating platform. If pushed toward the rim, its path will curve, rather than heading straight out along a radius.

When a long-range cannonball is fired towards the equator from a northern or southern latitude, it may land west of its intended longitude due to the rotation of the Earth. This is known as the Coriolis effect.

To understand why this happens, we need to consider two factors: the rotation of the Earth and the conservation of momentum. The Earth rotates from west to east, completing one full rotation in approximately 24 hours. As a result of this rotation, objects on the Earth's surface appear to be deflected to the right in the Northern Hemisphere and to the left in the Southern Hemisphere.

When a cannonball is fired, it inherits the eastward velocity of the Earth's surface at that particular latitude. However, as the cannonball moves towards the equator, it enters regions of the Earth's surface that are moving faster due to the Earth's rotation. This creates a relative velocity difference between the cannonball and the Earth's surface, resulting in a deflection caused by the Coriolis effect.

In the Northern Hemisphere, the Coriolis effect causes the cannonball to be deflected to the right of its intended path. This means that it will land west of its "intended" longitude. In the Southern Hemisphere, the deflection would be towards the left, causing the cannonball to land east of its "intended" longitude.

To calculate the amount of deflection caused by the Coriolis effect, you would need to consider the initial velocity of the cannonball, the latitude from which it was fired, and the time it took to reach its destination. Complex mathematical equations can be used to determine the precise deflection caused by the Coriolis effect.

It's worth noting that the Coriolis effect becomes more pronounced for long-range projectiles such as missiles or satellites traveling at high speeds over long distances. For shorter distances or slower speeds, the Coriolis effect is relatively minimal and can often be neglected.