Velocity Addition Calculator

Velocity addition determines how two motions combine into one observed velocity. In everyday mechanics, velocity is a vector: it has both magnitude and direction. The classical mode converts the two entered directions into their relative angle and applies the law of cosines. Its resultant magnitude is the square root of v1² + v2² + 2v1v2 cos(angle2 - angle1). Parallel velocities in the same direction add directly, opposing velocities subtract in magnitude, and perpendicular velocities form the familiar right-triangle result. This vector rule is appropriate for cars, aircraft, projectiles, flowing water, and nearly every ordinary engineering problem because their speeds are tiny compared with the speed of light. Angles are measured in degrees from a shared reference axis. Only the difference between the angles affects the magnitude. The calculator reports a nonnegative speed rather than a signed direction, so use component analysis if you also need the resultant angle or separate horizontal and vertical values. At speeds approaching light, direct addition would predict impossible speeds greater than light. Special relativity replaces the collinear sum with u = (v1 + v2) / (1 + v1v2 / c²), where c is exactly 299,792,458 meters per second. The denominator grows as the entered speeds increase, keeping the result below c whenever both input speeds are below c. The relativistic mode assumes both motions lie along the same line and in the positive direction. Its angle fields are retained for a consistent interface but do not alter that collinear calculation. Choose the model that matches the physical situation rather than simply choosing the more complicated formula. Classical vector addition preserves directional geometry and is effectively exact for low-speed applications. Relativistic addition is necessary when describing frames, particles, or signals moving at a significant fraction of light speed. Always express both speeds in meters per second before comparing or combining them. For opposite relativistic directions, represent the motion with signed velocities in a dedicated Lorentz transformation; this calculator accepts nonnegative magnitudes and is designed for same-direction relativistic examples. The displayed result is rounded for readability while the internal calculation uses JavaScript floating-point precision. It is suitable for education and quick analysis, but experimental uncertainty and significant figures should guide how many digits you report.