acceleration

Acceleration is the rate at which an object's velocity changes over time. It is a vector quantity, possessing both magnitude and direction. In one dimension, acceleration a is defined as a = Δv/Δt, where Δv is the change in velocity during a time interval Δt. Instantaneous acceleration is the limit as Δt approaches zero and is the derivative of velocity with respect to time, a = dv/dt.

Constant acceleration yields the standard kinematic relations for motion with initial velocity u and acceleration a: velocity v = u + a t; displacement s = u t + (1/2) a t^2; and v^2 = u^2 + 2 a s.

When motion involves changing direction, different forms of acceleration appear. Linear (translational) acceleration concerns motion along a straight line; angular acceleration α is the rate of change of angular velocity and is related to torque by τ = I α for a rotating rigid body. Centripetal acceleration a_c = v^2 / r points toward the center of the circular path.

Common examples include gravity, where near Earth's surface the acceleration of a freely falling body is approximately 9.81 m/s^2 downward, denoted g. Acceleration is related to force through Newton's second law, F = m a, where m is mass.

Units are meters per second squared (m/s^2). Observers in different frames may measure different coordinate accelerations, but the physical content is captured by the object's change in velocity over time. In relativity, the concept of proper acceleration generalizes this idea to the acceleration felt by an object, which can differ from coordinate acceleration in curved spacetime.