Velocity is a vector quantity describing an object's rate of change of position with respect to a chosen reference frame. Average velocity is displacement divided by elapsed time, while instantaneous velocity is the derivative of position with respect to time. If position is represented by the vector r, instantaneous velocity is v = dr/dt. The SI unit of velocity is the metre per second. Unlike speed, which describes the magnitude of motion, velocity includes direction.
An object travelling north at 20 m/s and an object travelling south at 20 m/s have equal speeds but different velocities. Velocity therefore depends on the reference frame and coordinate system used to describe the motion. A complete statement of velocity requires sufficient directional information to identify the vector.
The distinction between velocity and speed is fundamental. Speed is the magnitude of velocity and is therefore always non-negative in its ordinary physical interpretation, whereas a component of velocity can be positive, negative or zero depending on the selected coordinate direction. Average velocity depends on net displacement rather than total distance travelled.
If a runner travels around a track and finishes exactly where they started, their average velocity over the entire journey is zero even though their average speed is positive. Instantaneous velocity can likewise differ continuously from average velocity. This distinction becomes particularly important for journeys involving reversals of direction, circular motion, oscillations and complex trajectories. Treating distance and displacement, or speed and velocity, as interchangeable can produce incorrect conclusions about motion.
Velocity is closely connected to position and acceleration through calculus. The derivative of position with respect to time gives velocity, while the derivative of velocity with respect to time gives acceleration. Conversely, integrating velocity with respect to time gives displacement when the initial position is known. On a position-time graph, the gradient represents instantaneous velocity at a point and average velocity over an interval corresponds to the overall gradient between two points. On a velocity-time graph, the gradient represents acceleration and the signed area under the curve represents displacement. These graphical relationships allow motion to be analysed without necessarily solving explicit equations and are widely used in physics education, engineering measurements and experimental analysis.
Velocity always requires a reference frame. A passenger sitting inside a moving train has approximately zero velocity relative to the train carriage but a non-zero velocity relative to the ground. If two vehicles travel in opposite directions, their relative velocity is determined by subtracting one velocity vector from the other, which can produce a relative speed greater than either vehicle's individual speed.
Reference frames become especially important in mechanics because different observers can assign different positions and velocities while agreeing on the underlying physical laws in an appropriate classical frame. In special relativity, velocity transformations are modified so that the speed of light remains invariant for inertial observers, and ordinary Galilean addition of velocities no longer applies at relativistic speeds.
Velocity can be constant only when both its magnitude and direction remain constant. An object moving in a circle at constant speed therefore does not have constant velocity because its direction changes continuously. Similarly, an object can have a non-zero velocity while experiencing acceleration if its velocity is changing. Uniform rectilinear motion is the simplest case of constant velocity, in which an object covers equal displacements in equal time intervals along a straight line.
More complicated motion may involve variable velocity caused by changing forces, constraints or interactions. In projectile motion near Earth's surface, for example, the horizontal and vertical components of velocity can evolve differently when air resistance is neglected, with horizontal velocity remaining constant in the simplest model and vertical velocity changing under gravity.
Velocity is central to the mathematical analysis of forces and energy. Newton's laws use velocity indirectly through acceleration, while kinetic energy depends on the square of speed, which is the magnitude of velocity. Momentum is directly proportional to velocity through p = mv in classical mechanics, making velocity particularly important in collision analysis. The direction of velocity affects momentum even when speed is unchanged.
In fluid mechanics, velocity fields describe how fluid parcels move through space and time, allowing quantities such as flow rate, acceleration and pressure relationships to be studied. In electromagnetism and relativistic physics, velocity also influences how particles respond to fields and how observations transform between reference frames.
Practical measurements of velocity are performed using methods such as radar, Doppler measurements, GPS-derived position changes, optical tracking, wheel sensors and inertial systems. A measurement is only meaningful when its reference frame, temporal resolution and uncertainty are understood. GPS receivers, for example, estimate movement from changes in position and timing, while Doppler systems can infer radial velocity from frequency shifts. Experimental velocity measurements can contain errors due to sensor resolution, timing uncertainty, multipath effects, calibration problems or numerical differentiation of noisy position data. Differentiating noisy position measurements can amplify measurement noise, so filtering and appropriate estimation techniques are often required in engineering systems.
Common misconceptions include the belief that velocity and speed are interchangeable, that zero velocity necessarily means zero acceleration, and that constant speed implies constant velocity. None of these statements is generally correct. At the highest point of a vertically thrown object's trajectory, instantaneous vertical velocity can be zero while acceleration remains downward. An object travelling at constant speed around a circular path has changing velocity because its direction changes. Average velocity can also be zero after a journey with substantial movement if final and initial positions coincide. A precise understanding of velocity therefore requires attention to vector direction, reference frame, displacement, time interval and the distinction between instantaneous and average quantities.
