Velocity is a vector physical quantity that describes the rate at which an object's position changes with time. Average velocity is displacement divided by elapsed time, while instantaneous velocity is the derivative of position with respect to time. Unlike speed, velocity includes direction as well as magnitude. In one dimensional motion, direction can often be represented by a positive or negative sign, while in two or three dimensions velocity is represented by vector components.
The magnitude of instantaneous velocity is the instantaneous speed. Velocity is central to kinematics because acceleration is defined as the rate of change of velocity. An object can therefore accelerate even when its speed remains constant if its direction changes. The concept is fundamental to classical mechanics and has a relativistic generalisation for high speed motion.
Average velocity and average speed use different quantities and should not be confused. Average velocity equals net displacement divided by total elapsed time, whereas average speed equals total distance travelled divided by total elapsed time. If a runner travels around a track and finishes at the starting point, the displacement is zero, so the average velocity over the complete journey is zero. The total distance is nevertheless positive, so the average speed is positive.
This distinction becomes particularly important for journeys involving reversals of direction. An object can have a substantial average speed while having a small or zero average velocity. The difference follows directly from the distinction between displacement, which depends only on initial and final positions, and distance, which depends on the entire path taken.
Instantaneous velocity provides the direction and rate of position change at a specific moment. In calculus, velocity is the first derivative of position with respect to time. If position is represented by a vector function of time, velocity is also a vector and can be resolved into components along chosen coordinate axes. Acceleration is then the derivative of velocity with respect to time.
Integration reverses these relationships: integrating velocity over time gives displacement, while integrating acceleration over time gives the change in velocity. These mathematical relationships allow continuous motion to be described precisely. Graphs of position against time have slopes corresponding to velocity, while graphs of velocity against time have slopes corresponding to acceleration and areas corresponding to displacement.
Direction makes velocity fundamentally different from speed. Suppose a car travels north at 20 metres per second and another travels south at 20 metres per second. They have the same speed but different velocities because their directions differ. In vector notation, velocity can be decomposed into components such as east west and north south. Changes in any component represent changes in velocity and therefore contribute to acceleration. Circular motion demonstrates this especially clearly.
An object moving around a circle at constant speed continuously changes the direction of its velocity, so it has centripetal acceleration directed toward the centre of the circle. The velocity vector is tangent to the path at each instant, while the acceleration responsible for changing its direction points inward.
Velocity is reference frame dependent. A passenger walking forward inside a moving train has one velocity relative to the train and another relative to the ground. In classical mechanics, velocities between reference frames can be combined using Galilean velocity addition. If a person walks at 1 metre per second relative to a train moving at 20 metres per second relative to the ground, their ground relative velocity is approximately 21 metres per second when walking in the same direction. At relativistic speeds, this simple addition law fails and must be replaced by the relativistic velocity transformation. The relativistic formula ensures that no massive object or information signal is transformed into a speed exceeding the speed of light in vacuum.
Velocity is closely related to momentum. In classical mechanics, momentum equals mass multiplied by velocity, so momentum is a vector with the same direction as velocity for positive mass. Changing velocity therefore changes momentum, and net force can be expressed as the rate of change of momentum. In collisions, conservation of momentum allows velocities after an interaction to be calculated from initial conditions when the system is sufficiently isolated.
Velocity also determines kinetic energy in classical mechanics through the square of speed, although kinetic energy does not preserve direction because it is scalar. These relationships show why velocity is more informative than speed for analysing forces and interactions. Force changes momentum, and momentum depends directly on velocity.
Velocity can vary in several distinct ways. An object can increase or decrease its speed while maintaining direction, change direction while maintaining speed, or change both speed and direction simultaneously. Acceleration describes all such changes. In projectile motion near Earth's surface, horizontal velocity can remain approximately constant while vertical velocity changes because of gravity, neglecting air resistance. In circular motion, the speed can remain constant while the velocity direction changes continuously. In oscillatory motion, velocity changes in both magnitude and direction as the object moves through equilibrium and reverses direction. These examples demonstrate why velocity is the appropriate quantity for describing motion dynamically rather than relying solely on speed.
At relativistic speeds, velocity remains a meaningful physical quantity but its mathematical treatment changes. Different inertial observers do not simply add velocities using ordinary arithmetic. The relativistic velocity transformation follows from Lorentz transformations and preserves the invariant speed of light. Proper velocity, four velocity and coordinate velocity are additional specialised concepts used in relativistic physics, with different mathematical meanings from the ordinary velocity taught in introductory mechanics.
In most terrestrial situations, these distinctions are unnecessary because speeds are tiny compared with the speed of light. Nevertheless, the relativistic framework becomes essential for particle accelerators, high energy astrophysics and precision satellite navigation. Velocity is therefore a basic concept whose elementary definition remains useful across physics while its deeper mathematical treatment depends on the physical regime.
