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In the previous sections, we have derived an equation that describes the motion of a particle, as observed from a coordinate system that is undergoing acceleration and rotation relative to a “fixed” coordinate system.  The equation is as follows:

where F is the total force acting on the particle, ω is the angular velocity vector of rotation (assumed to be constant), m is the particle mass, A is the translational acceleration of the coordinate system, and a, v and r are acceleration, velocity and position respectively.  The primes on these last three symbols indicate that they are measured relative to the rotating coordinate system.

In this section, our aim is to derive the equations of motion for a projectile, as observed from a coordinate system located on the plant Earth.  By projectile, we mean the motion of a body or particle under the influence of gravity.  Examples could be a bullet fired from a gun, or a thrown baseball or cricket ball.

Since the earth is rotating, a coordinate system located on earth does not constitute an inertial frame of reference, though of course we often approximate such frames of reference as inertial.  The aim here, however, is to see how we proceed – and what results – when we relax the assumption that a coordinate system on Earth is inertial.  In order to investigate the dynamics of a projectile under these conditions, we need to apply equation (A) to the motion of the projectile.

We begin by defining the coordinate system with which we are going to work.  This coordinate system is illustrated in the following figure.

This figure shows a coordinate system, origin O’, located at a point at latitude λ on the earth’s surface.  The x’ axis points east, and hence points into the plane of the above figure.  As the latitude λ varies, so the orientation of the coordinate system O’ varies relative to the angular velocity vector ω. 

Our first task is to examine the term F – mA in equation (A).  To do this, we note that A is the acceleration of the coordinate system O’.  This acceleration is just the centripetal acceleration due to the rotation of O’ on the earth’s surface.  This has magnitude

and is directed horizontally towards the axis of rotation of the earth.  The force F is just the force due to gravity on the projectile, and this points towards the centre of the earth.  Thus, the term F – mA consists of the vector difference of the two forces mg and F_C shown in the following diagram:

Now, it is possible to compute the resultant force, which would involve consideration of a vector triangle of the form shown in the following diagram:

The result would be an effective g’ that acts in a slightly different direction than g.  However, it is not difficult to show that the angle ε is no more than about a tenth of a degree.  This arises because the centripetal acceleration is much smaller than the acceleration due to gravity.  This in turn results from the low angular velocity of the earth, which is equal to one rotation per day, or about 7.3 10^-5 radians per second.  For this reason, we will neglect the difference between g’ and g, and hence from now on we will take

For similar reasons, the magnitude of the centrifugal force

is small compared with the other terms in equation (A).  This is because the magnitude of the centrifugal force is quadratic in the angular velocity ω. 

The equation of motion of a projectile in the rotating frame therefore reduces to

We proceed by splitting this equation into its three Cartesian components.  To do this we note that in the coordinate system O’, the angular velocity vector can be written as

We also note that

with similar expressions for r’ and the second derivative of r’.

The equation of motion then separates into the following three differential equations:

These equations can be integrated immediately with respect to time.  However, before doing this, let us specify the initial conditions for the motion of our projectile.  We specify the initial position vector of the projectile as

and the initial velocity vector as

In this expression, the dot is used to represent differentiation with respect to time.  The integrated equations of motion become:

The second and third of these can be substituted into equation (B) to give

In this expression, terms in ω² are neglected because ω itself is small (see above).  This expression can be integrated twice to give

This expression can be substituted into the expressions for the derivatives of y’ and z’ above.  Neglecting terms in ω² and integrating then gives

These last three equations define the motion of a projectile in a frame of reference located on the earth’s surface, taking into account the rotation of the earth.  In the next section, some examples of the application of these equations are presented.

Examples of Projectile Motion