{dt} = \frac{d^2\mathbf{x}}{dt^2} | dimension = wikidata
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the orientation of the net force acting on that object. The magnitude of an object's acceleration, as described by Newton's second law, is the combined effect of two causes:
The SI unit for acceleration is metre per second squared (, ).
For example, when a vehicle starts from a (zero velocity, in an inertial frame of reference) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the vehicle turns, an acceleration occurs toward the new direction and changes its motion vector. The acceleration of the vehicle in its current direction of motion is called a linear (or tangential during ) acceleration, the reaction to which the passengers on board experience as a force pushing them back into their seats. When changing direction, the effecting acceleration is called radial (or centripetal during circular motions) acceleration, the reaction to which the passengers experience as a centrifugal force. If the speed of the vehicle decreases, this is an acceleration in the opposite direction of the velocity vector (mathematically a negative number, if the movement is unidimensional and the velocity is positive), sometimes called deceleration Extract of page 39 Extract of page 36 or retardation, and passengers experience the reaction to deceleration as an force pushing them forward. Such negative accelerations are often achieved by retrorocket burning in spacecraft. Both acceleration and deceleration are treated the same, as they are both changes in velocity. Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their relative (differential) velocity are neutralised in reference to the acceleration due to change in speed.
(Here and elsewhere, if motion is in a straight line, Euclidean vector quantities can be substituted by scalars in the equations.)
By the fundamental theorem of calculus, it can be seen that the integral of the acceleration function is the velocity function ; that is, the area under the curve of an acceleration vs. time ( vs. ) graph corresponds to the change of velocity.
Likewise, the integral of the jerk function , the derivative of the acceleration function, can be used to find the change of acceleration at a certain time:
Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an accelerometer.
In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law): where is the net force acting on the body, is the mass of the body, and is the center-of-mass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large.
where is the unit (inward) normal vector to the particle's trajectory (also called the principal normal), and is its instantaneous radius of curvature based upon the osculating circle at time . The components
Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas.
A frequently cited example of uniform acceleration is that of an object in free fall in a uniform gravitational field. The acceleration of a falling body in the absence of resistances to motion is dependent only on the gravitational field strength standard gravity (also called acceleration due to gravity). By Newton's second law the force acting on a body is given by:
Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating the displacement, initial and time-dependent velocity, and acceleration to the time elapsed:
where
In particular, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations. As Galileo showed, the net result is parabolic motion, which describes, e.g., the trajectory of a projectile in vacuum near the surface of Earth.
As usual in rotations, the speed of a particle may be expressed as an angular velocity with respect to a point at the distance as
Thus
This acceleration and the mass of the particle determine the necessary centripetal force, directed toward the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion. The so-called 'centrifugal force', appearing to act outward on the body, is a so-called pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum, a vector tangent to the circle of motion.
In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to the principal normal, which directs to the center of the osculating circle, that determines the radius for the centripetal acceleration. The tangential component is given by the angular acceleration , i.e., the rate of change of the angular speed times the radius . That is,
The sign of the tangential component of the acceleration is determined by the sign of the angular acceleration (), and the tangent is always directed at right angles to the radius vector.
As speeds approach that of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached; an object with mass can approach this speed asymptotically, but never reach it.
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