accelerations

Acceleration is the rate at which an object's velocity changes over time. It is a vector quantity, having both magnitude and direction, and is typically denoted by a. Instantaneous acceleration a(t) is the derivative of velocity with respect to time, a(t) = dv/dt. The average acceleration over a time interval Δt is Δv/Δt. The SI unit is meters per second squared (m/s^2).

In one dimension, acceleration is the time derivative of speed or velocity. In multiple dimensions, it is a vector with components along each axis. For a particle with position r(t), acceleration is a = d^2r/dt^2. If acceleration is constant, velocity changes linearly with time and displacement follows s = ut + 1/2 a t^2, v = u + a t, where u is the initial velocity.

Common types include linear and angular acceleration. Angular acceleration α = dω/dt, where ω is angular velocity; for constant α, ω = ω0 + α t and θ = ω0 t + 1/2 α t^2, with θ the angular displacement. In circular motion, centripetal (radial) acceleration a_c = v^2/r points toward the center, and any tangential acceleration a_t is attributable to changing speed.

Acceleration relates to forces through Newton's second law: F = ma for linear motion, and τ = Iα for rotation, where I is the moment of inertia. In relativity, acceleration can be described in terms of proper acceleration measured by an accelerometer, and gravitational effects can be described as either acceleration relative to frames or as spacetime curvature. Earth’s surface provides about 9.81 m/s^2 downward as a standard reference.

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