Acceleration is the rate at which velocity changes with time. Because velocity is a vector quantity, acceleration can result from a change in speed, a change in direction, or simultaneous changes in both. Average acceleration is calculated as the change in velocity divided by the corresponding change in time, while instantaneous acceleration is the derivative of velocity with respect to time.
In one-dimensional motion this can often be written as a = Δv divided by Δt for average acceleration, or a = dv/dt for instantaneous acceleration. The SI unit of acceleration is the metre per second squared, written m/s². The unit expresses how much velocity changes for each second that passes. Acceleration is therefore not synonymous with increasing speed; an object travelling at constant speed around a curved path is accelerating because its velocity continually changes direction.
Acceleration is a vector and consequently has both magnitude and direction. In Cartesian coordinates, acceleration can be resolved into components such as aₓ, aᵧ and a_z, allowing complicated motion to be analysed independently along different axes. For straight-line motion, the acceleration direction can be compared directly with the velocity direction. If acceleration and velocity point in the same direction, the object's speed increases. If they point in opposite directions, speed decreases.
If they are perpendicular, the instantaneous speed may remain unchanged while the direction of velocity changes. In circular motion, acceleration can be divided into tangential and centripetal components. Tangential acceleration changes speed, while centripetal acceleration changes direction. This vector nature is why statements such as an object having an acceleration of 5 m/s² are incomplete unless the relevant direction or coordinate convention is understood.
Newton's second law connects acceleration to force and mass through F = ma in classical mechanics. For a given net force, an object's acceleration is inversely proportional to its inertial mass, while for a given mass, acceleration is proportional to the net force. Importantly, the force in this equation is the net external force rather than simply any individual force acting on the object.
A book resting on a table, for example, can have gravitational and normal forces acting on it while its net force and acceleration are both zero. If the forces cease to balance, the resulting net force produces acceleration. This relationship provides one of the central bridges between the mathematical description of motion and the physical causes of motion. It also explains why increasing the applied force does not necessarily increase acceleration if another force simultaneously increases and offsets it.
Constant acceleration produces several important kinematic relationships. If acceleration is constant, final velocity can be found from v = u + at, where u is initial velocity, v is final velocity and t is elapsed time. Displacement can be calculated using s = ut + ½at², and velocity and displacement can be related through v² = u² + 2as. These equations are derived from the definitions of velocity and acceleration under the assumption of constant acceleration.
They are particularly useful for idealised problems involving falling objects, vehicles, projectiles and simple mechanical systems. When acceleration varies with time, these constant-acceleration equations generally cannot be applied directly, and calculus or numerical methods may be required. Graphically, acceleration corresponds to the gradient of a velocity-time graph, while velocity corresponds to the gradient of a displacement-time graph.
Gravity provides one of the most familiar examples of acceleration. An object undergoing ideal free fall near Earth's surface has an acceleration of approximately 9.8 m/s² downward, regardless of its mass when air resistance is neglected. A falling object therefore gains approximately 9.8 m/s of downward velocity during each second of idealised motion. The acceleration remains approximately constant only over relatively modest changes in height and under conditions where air resistance is negligible.
As an object falls through an atmosphere, drag increases with speed and can eventually balance the gravitational force, producing terminal velocity and therefore approximately zero net acceleration. This illustrates why gravitational acceleration and the actual acceleration of a falling object are related but not always numerically identical in real conditions.
Acceleration is also fundamental to rotational and curved motion. An object moving in a circular path at constant speed experiences centripetal acceleration directed towards the centre of the circle, with magnitude a_c = v²/r, where v is speed and r is radius. The object is accelerating despite its speed remaining constant because its velocity direction changes continuously. If the object's speed changes as it travels around the circle, tangential acceleration must also be present.
In more advanced mechanics, acceleration can be decomposed into normal and tangential components relative to a curved trajectory. This approach is useful in vehicle dynamics, robotics, aerospace engineering, machine design and analysis of motion along arbitrary paths. The distinction between translational acceleration and angular acceleration is also important: angular acceleration describes the rate of change of angular velocity and is measured in radians per second squared.
Acceleration can be observed in everyday systems including cars, lifts, aircraft, amusement rides, machinery and electronic motion sensors. Accelerometers measure proper acceleration or related quantities and are used in smartphones, navigation systems, industrial monitoring, vehicle safety systems and inertial measurement units. A sensor mounted in a stationary object on Earth's surface can register acceleration associated with the supporting force because an accelerometer does not simply report the coordinate acceleration of an object in the same way a position tracker does.
This distinction becomes particularly important in relativistic physics and inertial navigation. In engineering, acceleration limits may be imposed to protect occupants, structures, equipment or manufactured components from excessive loads. Sudden acceleration changes can also be important because the derivative of acceleration, sometimes called jerk, can affect mechanical comfort and structural stress.
Several common misconceptions about acceleration arise from treating it as a synonym for speeding up. A vehicle braking from 30 m/s to 10 m/s has acceleration opposite to its direction of travel, while a vehicle maintaining constant speed around a bend has non-zero acceleration. Zero acceleration means constant velocity, not necessarily zero velocity. Similarly, an object can have zero velocity at an instant while having substantial acceleration, as occurs at the highest point of an ideal vertical throw.
Acceleration must therefore be analysed from velocity rather than speed alone. In advanced contexts, coordinate acceleration and proper acceleration can differ, especially in relativity. Despite these distinctions, the central definition remains precise: acceleration quantifies how velocity changes with time, making it one of the fundamental kinematic quantities used to describe motion.
