Coriolis effect is the acceleration (and therefore deflection) of an object in motion with respect to a non-inertial frame of reference (in this page we will focus on a non-inertial frame of reference on Earth’s surface). The inertial force causing this acceleration is called the Coriolis force.
In this page we derived an equation that relates the acceleration of a particle measured by two different frames of reference: the first one (O) is at rest (and therefore is an inertial frame of reference) and the second one (O’) is a non-inertial frame of reference rotating at constant angular velocity ω.
This equation reads:
where the prime symbol corresponds to quantities measured by the rotating non-inertial frame of reference (O’). The other quantities are measured by the inertial frame of reference at rest (O).
The last two terms at the right side of the equation are called, respectively, Coriolis acceleration and centrifugal acceleration.
The fact that the acceleration of an object depends on the observer means that the same object will move differently depending on the motion of the observer himself.
Imagine the simplest case where an object is at rest with respect to O, the frame of reference at rest. For him, this object will have zero acceleration.
However, from the point of view of the rotating frame of reference O’, this same object will have an acceleration given by:
and therefore with respect to O’ it will not be at rest.
In other words, since O’ is not aware of his own rotational motion, he comes to the conclusion that it is the object he is observing that is in motion.
Motion relative to Earth
The Earth rotates anticlockwise about its axis at (approximately) constant angular velocity ω; therefore, any observer on its surface is a non-inertial frame of reference O’. So when such an observer measures the acceleration of any object, this measurement will include the effect of the Earth’s own rotation. This effect is given by both the Coriolis and the centrifugal acceleration.
In this page we will examine the effect the Coriolis acceleration, which is given by:
where ω is the angular velocity of the Earth and v’ is the velocity of the moving object as measured by the observer O’ on the surface of the Earth.
The angular velocity ω is a vector whose magnitude is the angular speed ω and its direction is upwards along the rotational axis of Earth.
The cross product of the two vectors is a third vector that is perpendicular to both ω and v’. Its direction is given by the right hand rule.
In order to understand the Coriolis effect, we are going to analyze the different outcomes of the following “thought experiment”: a projectile is launched from a point on Earth’s surface pointing directly towards a target located at different latitude. Will the projectile hit the target? If not, why not?
We will analyze the projectile motion from the point of view of both the rotating frame of reference O’ and the frame of reference at rest O.
There are four different scenarios to consider: the projectile is launched from the Northern Hemisphere, northward and southward, and from the Southern hemisphere (also norward and southward).
-Northern Hemisphere
As you can see in this page, the farther a point on Earth’s surface is from the Equator, the smaller its linear speed is. On the other hand, from the point of view of an observer at rest O, the velocity of the projectile has two components: the velocity (v’) at which it is launched from Earth (green arrow in the figure below) and the horizontal velocity of the point on Earth’s surface from where it was fired (grey arrow in the figure below). Figure (a) shows the projectile fired northward and (b) southward.
When the projectile is fired northward, from the point of view of the observer at rest O, by the time the projectile reaches the latitude where the target is (red dot in the figure below), the target will have moved from its initial position A to point B. But, in the same amount of time, the projectile will move horizontally eastward from point 1 to point 2.
The horizontal distance traveled by the projectile is greater than the displacement of the target (because its horizontal speed is greater as well). So the projectile misses the target because the latter lags behind.
Conversely, when the projectile is fired southward (b), the horizontal distance it travels is smaller than the distance traveled by the target. So it is also a miss, but this time it’s the projectile that lags behind the target.
From the point of view of the observer O’ on Earth’s surface, in both cases the projectile misses the target because its trajectory is deflected toward the right with respect to the direction of v’. But a curved path means the object is accelerating. It is the Coriolis acceleration that is continually changing the direction of the velocity.
In the 3D model below it is shown how the projectile motion is seen by an observer O’ on Earth. You can interact with the model to see it from different angles. You can see how to find the cross product of two vectors here.
-Southern Hemisphere
When the projectile is fired from the Southern Hemisphere (figure below), the situation is reversed. From the point of view of an observer at rest O, when it is launched northward (a), it lags behind the target, because this time the target is closer to the Equator and therefore it moves eastward faster that the projectile. On the other hand, when it is launched southward (b), it surpasses the target for the opposite reason.
From the point of view of the observer O’ on Earth’s surface, in both cases the projectile misses the target, because its trajectory is deflected toward the left due to the Coriolis acceleration. In the 3D model above you can see what happens from the point of view of O’ when a projectile is launched from the Southern Hemisphere.
Lastly, Newton’s second law states that acceleration is always caused by a force. The force causing the Coriolis acceleration is the Coriolis force. It is a inertial force because it is only “seen” by the non-inertial frame of reference O’ on Earth’s surface. For the inertial frame of reference O this force doesn’t exist.
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