Coriolis Force Simulator

Simulation summary will appear here.

Real-World Phenomenon

Objects that move across Earth’s surface do not travel in perfect straight lines when viewed from the ground. Because the planet rotates, observers in this non‑inertial frame perceive a sideways acceleration known as the Coriolis force. The simulator above depicts an idealized projectile launched near Earth’s surface. The blue dot represents the mass in the rotating frame, and the red curve traces its path. A bar beneath the canvas tracks kinetic energy and the slight numerical error that accumulates in the integration. This visual approach clarifies how the Coriolis term bends trajectories without performing work.

Variables and Assumptions

The model assumes a flat tangent plane at latitude $\phi$ and neglects centrifugal effects, air resistance, and changes in gravity with altitude. Coordinates are east ($x$) and north ($y$). The object of mass $m$ begins at the origin with speed $v$ and heading angle $\theta$ measured counter‑clockwise from east. Earth’s rotation rate is $\Omega =7.292115\times {10}^{-} rad∕s$. The Coriolis acceleration in the horizontal plane is

$\mathrm{a_x}=-2\Omega \mathrm{v_y}\sin \phi$,   $\mathrm{a_y}=2\Omega \mathrm{v_x}\sin \phi$

which ensures the acceleration is perpendicular to velocity, conserving kinetic energy in theory.

Core Equations

The simulator integrates the kinematic system

$\mathrm{\dot{}x}=\mathrm{v_x}$, $\mathrm{\dot{}y}=\mathrm{v_y}$, $\mathrm{\dot{}v_x}=-2\Omega \mathrm{v_y}\sin \phi$, $\mathrm{\dot{}v_y}=2\Omega \mathrm{v_x}\sin \phi$

derived from the Coriolis force expression $F_c=2mv\Omega \sin \phi$. The kinetic energy is $\mathrm{E_k}=\frac{1}{2}m({\mathrm{v_x}}^2+{\mathrm{v_y}}^2)$.

Numerical Scheme

An explicit fourth‑order Runge–Kutta (RK4) integrator advances the state by time step $\mathrm{\Delta t}$. RK4 balances accuracy and performance for smooth problems. The algorithm samples intermediate slopes to cancel lower‑order truncation errors, preserving energy to within machine precision for modest step sizes. The user may modify $\mathrm{\Delta t}$ to explore stability limits. If the step is too large, the energy bar’s red segment indicates growing numerical drift.

Worked Example

Consider a 1 kg mass fired eastward at 100 m/s from latitude 45°. With $\mathrm{\Delta t}=0.1 s$ and total time 100 s, the theoretical deflection after 100 s is roughly 7.3 km southward. Running the simulator produces a red trajectory curving gently toward the right (south). The kinetic energy bar remains nearly constant, showing that Coriolis forces do no work.

Comparison Table

Scenario	Latitude φ	Speed (m/s)	Deflection after 100 s (km)
Baseline	45°	100	7.3
High latitude	60°	100	10.1
Slower speed	45°	50	3.6

How to Read the Animation

The blue dot shows the mass viewed from Earth. The red path indicates accumulated deflection. The kinetic energy bar (blue) should stay constant; any red segment highlights numerical error $\mathrm{\Delta E}$. The text summary beneath the controls mirrors the canvas description for screen‑reader users.

Limitations

The model ignores curvature of Earth, altitude changes, air drag, and centrifugal adjustments. It assumes small regions where the tangent plane approximation holds. Real projectiles experience additional forces from gravity gradients and atmospheric resistance, so actual paths differ slightly. Energy conservation checks only evaluate Coriolis work and do not account for integrator truncation beyond kinetic energy.

Suggested Extensions

Future versions could include vertical motion with gravity, visualization on a sphere, or coupling to atmospheric pressure gradients for geophysical flows. Incorporating a symplectic integrator would further reduce long‑term energy drift.

References

Gill, A. E. Atmosphere–Ocean Dynamics. Academic Press, 1982.

Holton, J. R. Introduction to Dynamic Meteorology. Elsevier, 2004.

Related tools: Centripetal Force Calculator, Ballistic Trajectory Simulator, Foucault Pendulum Precession.