Two-dimensional collision calculator

You supply the two masses and the Cartesian components of each body's initial velocity; the engine applies conservation of momentum separately on the x-axis and the y-axis and returns v_fx equals open parenthesis m1 times v1x plus m2 times v2x close parenthesis divided by open parenthesis m1 plus m2 close parenthesis, and the same form for v_fy with the y-components.

What a perfectly inelastic 2-D collision means and what the response contains

That identity yields the closed forms v_fx equals open parenthesis m1 v1x plus m2 v2x close parenthesis over open parenthesis m1 plus m2 close parenthesis and v_fy equals open parenthesis m1 v1y plus m2 v2y close parenthesis over open parenthesis m1 plus m2 close parenthesis. The response returns the normalized inputs m1, v1x, v1y, m2, v2x, and v2y, the total_mass m1 plus m2, the total_momentum_x and total_momentum_y components, the final_velocity_x and final_velocity_y of the combined body, the final_speed as the Euclidean magnitude of that vector, the final_angle_deg measured from the positive x-axis with atan2, the precision used, a method tag perfectly_inelastic_collision_2d, a short formula reminder, and a unit hint m/s when masses are in kilograms and velocities in meters per second. There is no multi-body contact simulation, no coefficient of restitution path, and no rotation or friction model: the engine either evaluates the momentum quotients inside the supported numeric range or rejects the input with invalid_input. Rejected cases include missing masses or velocity components, non-finite values, magnitudes above the documented ceilings, negative mass, and any case where total mass is zero or numerically zero, because division by zero leaves no defined common final velocity.

How masses and velocity components are read and how v_f stays reproducible

Zero mass for one body is allowed when the other mass is positive, which simply reduces to the remaining body keeping its velocity vector. Both masses zero is rejected because total mass must not be zero. Velocity components may be positive, negative, or zero to encode direction in the plane; the tool does not convert unit systems for you, so keep masses consistent with each other and velocities in a matching length-per-time unit. Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces and non-numeric tokens are rejected. The preferred fields are m1 and m2 for the two masses and v1x, v1y, v2x, and v2y for the four initial velocity components, but aliases mass1, M1, and mass_1 for m1, mass2, M2, and mass_2 for m2, and vx1, velocity1_x, u1x for v1x plus the matching aliases for the other three components are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Masses must be finite and non-negative: a negative mass fails with invalid_input and a message that names the field, so a bad cell does not become a silent nonsense result. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for the numeric outputs, defaulting to ten and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the planar inelastic-collision formulas yourself.

Where two-dimensional inelastic collisions fit in study, design, and software

Product and education tools use the endpoint when a learner types two masses and four velocity components and the UI must show the combined velocity after a perfectly inelastic planar crash. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with the one-dimensional inelastic helper when the motion is collinear, with elastic-collision helpers when bounce speeds are needed, and with other math endpoints when you are building a small lab of free, deterministic physics calculators for two-dimensional mechanics.