Relative velocity 2D calculator

Given the Cartesian components of body 1 and body 2, this relative velocity 2D calculator forms v_rel equals v1 minus v2, then reports the relative components, the Euclidean magnitude, and the polar direction from the positive x-axis using atan2. Every component must be a finite number; non-numeric values fail with invalid_input and are not billed.

What 2D relative velocity means and what the response contains

In introductory mechanics and planar kinematics, the velocity of object A relative to object B is the vector that an observer riding with B would assign to A. The direction is the angle of the relative vector measured counterclockwise from the positive horizontal axis, returned both in radians and in degrees so worksheets that prefer either convention can copy a single field without converting by hand. Zero relative velocity is allowed and yields magnitude zero with direction zero from atan2 of zero and zero, matching common library behavior. When both velocities are already resolved into Cartesian components in a shared inertial frame, that relative velocity is simply the component-wise difference: relative_vx equals v1x minus v2x and relative_vy equals v1y minus v2y. The magnitude of that difference is the Euclidean length sqrt of relative_vx squared plus relative_vy squared, which is the speed at which A approaches or recedes from B along the connecting relative path when positions are also known. This endpoint does not integrate trajectories, does not apply Coriolis corrections, and does not change frames that are rotating; it answers only the static vector difference of two finite planar velocities. The success payload echoes the four input components after stable rounding, then exposes relative_vx, relative_vy, magnitude, direction_deg, direction_rad, a short formula string, a method tag vector_difference_2d, and a unit hint that any consistent length-over-time unit may be used when inputs share that unit. Pair this tool with velocity-components helpers when you only know speed and angle, or with collision endpoints when you need final velocities after an impact rather than the pre-impact relative vector alone.

Input contract, numeric validation, and error behaviour

Magnitudes above the documented absolute ceiling are rejected to keep intermediate products finite. Failed validation is not charged. Keep units consistent yourself: the calculator does not convert kilometres per hour into metres per second; it only subtracts numbers you already expressed in one coherent system. POST a JSON object with four required fields: v1x and v1y for the first velocity, and v2x and v2y for the second. Each field may be a JSON number or a decimal numeric string, including optional leading signs and scientific notation that parse as finite values. Preferred names are the canonical DOM-safe identifiers; aliases vx1, vy1, vx2, vy2 and u_x, u_y, w_x, w_y are accepted so spreadsheets and typed forms can reuse common textbook symbols without renaming columns. Missing fields, empty strings, NaN, Infinity, and non-numeric tokens such as letters or mixed units fail with invalid_input and a message that names the offending field, so a bad cell never becomes a silent zero. The free browser path and the paid API path share the same pure module, which means a worksheet example you verify on this page will match an automated pipeline that posts the same four numbers at $0.002 per successful item.

Where a relative velocity 2D calculator earns its keep

Teachers pre-generate answer keys for modest component sets and reject non-numeric student paste with a clear invalid_input signal instead of a mysterious empty chart. Product and education tools call the endpoint when a learner types four boxes and the UI must show metres per second of closing speed plus a direction badge.