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Compare the same free puck in a stationary laboratory frame and a frame rotating with a frictionless horizontal platform. The laboratory path is straight; a curved path in the rotating view comes from the changing frame of reference.

Release conditions

Set constant angular velocity ω from −4 to 4 rad/s, initial speed v₀ from 0 to 6 m/s, throw angle α from −180 to 180°, and start radius r₀ from 0 to 1.2 m. Zero values are valid. The platform radius is 1.45 m. All calculations run in your browser.

Both coordinate systems are aligned when the puck is released at t = 0, at (r₀, 0). Viewed from above, positive ω means counterclockwise rotation. The initial speed v₀ is relative to the platform. The angle α is measured counterclockwise from the outward radial +x direction: 0° points outward, 90° is counterclockwise tangential, and −90° is clockwise tangential.

Use

Select Calculate to prepare both complete paths. The time field and slider select the same instant in both views. Play moves the probe in real time until the calculated end; Pause stops it. Invalid time input hides current markers and values without changing the paths. Changing a release condition clears both paths and pauses playback. Reset result keeps the release conditions.

The app pauses when the tool is hidden or you navigate away. Returning within the app preserves the paused result. Reloading restores only validated release conditions and the expanded-details preference, then requires a new calculation.

Examples

• Radially outward uses ω = 1.2 rad/s, v₀ = 2.5 m/s, α = 0° and r₀ = 0.6 m. Reversing ω mirrors the y components of this radial example.
• With a stationary platform, ω = 0 and both views coincide exactly.
• Release with platform sets v₀ = 0. The puck keeps the platform's tangential laboratory velocity ωr₀; zero relative release speed does not normally mean laboratory rest.
• At rest in laboratory uses ω = 1.2 rad/s, v₀ = 0.72 m/s, α = −90° and r₀ = 0.6 m. The throw cancels the platform's tangential velocity.
• Throw from center sets r₀ = 0, so there is no initial tangential platform velocity.

Positions and velocities

Write I for laboratory coordinates, R for rotating coordinates, and J(x, y) = (−y, x). The laboratory velocity is vᴵ = (v₀ cos α, v₀ sin α + ωr₀), and the position is rᴵ(t) = (r₀, 0) + vᴵt. Rotation into the moving frame gives rᴿ = R(−ωt)rᴵ, where R is the usual counterclockwise rotation matrix. Thus xᴿ = cos(ωt)xᴵ + sin(ωt)yᴵ and yᴿ = −sin(ωt)xᴵ + cos(ωt)yᴵ.

The instantaneous rotating-frame velocity is vᴿ = R(−ωt)vᴵ − ωJrᴿ. The Coriolis acceleration is aCor = −2ωJvᴿ, the centrifugal acceleration is aCent = ω²rᴿ, and the total rotating-frame acceleration is aᴿ = aCor + aCent. The Coriolis term uses the current rotating-frame velocity, not a fixed initial speed. These accelerations arise from the rotating coordinates; there is no horizontal external force on the laboratory puck.

Open the component details for positions, velocities and the two acceleration terms. Both graphs use the same scale, with x and y in meters and coordinate limits ±1.65 m. The platform spokes and release mark rotate counterclockwise for positive ω in the laboratory; they remain fixed in the rotating frame.

End of the path

Both paths stop at the first contact with the 1.45 m platform rim, or after an observation window of 5 s if no contact occurs sooner. The contact time is the analytic intersection of the straight laboratory path with the circle. A puck at rest in the laboratory has no rim-contact event. The model does not continue a collision or a fall beyond the rim.

The traces come from analytic positions with at least 100 intervals and at most 0.02 rad of platform rotation per interval. Each contains at most 1001 points. The time probe is evaluated separately and is not limited to those stored points. Displayed values are rounded, with scientific notation for small nonzero values.

Assumptions and source

The platform is horizontal and frictionless, with constant angular velocity and no horizontal external force on the puck. Friction, Earth rotation, vertical ballistics and changes in angular velocity are outside the model.

Douglas Cline, §12.8 Coriolis Force, equations 12.41 and 12.42 and the free-puck example, supplies the rotating-frame acceleration relations. The coordinate solution, initial-velocity conversion and finite path bounds here are derived for this tool.