1. **Problem Statement:** We are given the full equations of motion involving forces such as Coriolis Force (CF), Pressure Gradient Force (PGF), gravity (g), and Friction (F). The goal is to understand the components of acceleration in the x and y directions and how these forces interact. 2. **Equations and Definitions:** The full equation of motion is: $$RA + dFA = PGF + GRF + FRF$$ or simplified as: $$RA = CF + PGF + g + F$$ where: - $RA$ is the resultant acceleration, - $CF$ is the Coriolis Force, - $PGF$ is the Pressure Gradient Force, - $g$ is gravity, - $F$ is friction. 3. **Coordinate System and Variables:** We consider the coordinates $(x,y,z)$ and velocity components $(u,v,w)$. 4. **Equations of Motion in x and y directions:** $$\frac{du}{dt} = 2\Omega \sin \phi \ v - 2\Omega \cos \phi \ w - \frac{1}{\rho} \frac{dp}{dx} + F_x$$ $$\frac{dv}{dt} = -2\Omega \sin \phi \ u - \frac{1}{\rho} \frac{dp}{dy} + F_y$$ where $\Omega$ is the Earth's rotation rate, $\phi$ is latitude, $\rho$ is density, and $p$ is pressure. 5. **Geostrophic Flow Approximation:** Define the Coriolis parameter: $$f = 2\Omega \sin \phi$$ Using this, the pressure gradient terms relate to velocity as: $$\frac{dp}{dx} = -f \rho v$$ $$\frac{dp}{dy} = f \rho u$$ which implies: $$f_r = \frac{1}{\rho} \frac{dp}{dx} = -f v$$ $$f_u = -\frac{1}{\rho} \frac{dp}{dy} = -f u$$ 6. **Interpretation of Coriolis Force:** The Coriolis force arises due to Earth's rotation and causes an apparent deflection of moving air to the right in the Northern Hemisphere. 7. **Summary:** The acceleration in the x-direction depends on the Coriolis force term $2\Omega \sin \phi v$, vertical motion term $-2\Omega \cos \phi w$, pressure gradient force $-\frac{1}{\rho} \frac{dp}{dx}$, and friction $F_x$. Similarly, the y-direction acceleration depends on $-2\Omega \sin \phi u$, pressure gradient force $-\frac{1}{\rho} \frac{dp}{dy}$, and friction $F_y$. This set of equations models atmospheric motion including geostrophic wind balance and Coriolis effects. **Final note:** These equations are fundamental in meteorology and oceanography for understanding wind and current patterns.