This article is about acceleration in physics. For other uses, see Acceleration (disambiguation). "Accelerate" redirects here. For other uses, see Accelerate (disambiguation).
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are 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 net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force;
- that object's mass, depending on the materials out of which it is made — magnitude is inversely proportional to the object's mass.The SI unit for acceleration is metre per second squared (m⋅s−2, ).
For example, when a vehicle starts from a standstill (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 circular motions) 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, if the movement is unidimensional and the velocity is positive), sometimes called deceleration or retardation, and passengers experience the reaction to deceleration as an inertialforce pushing them forward. Such negative accelerations are often achieved by retrorocketburning 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 neutralized in reference to the acceleration due to change in speed.
Definition and properties
An object's average acceleration over a period of time is its change in velocity, , divided by the duration of the period, . Mathematically,
Instantaneous acceleration
Instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimalinterval of time. In the terms of calculus, instantaneous acceleration is the derivative of the velocity vector with respect to time:
As acceleration is defined as the derivative of velocity, v, with respect to time t and velocity is defined as the derivative of position, x, with respect to time, acceleration can be thought of as the second derivative of x with respect to t:
(Here and elsewhere, if motion is in a straight line, 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 a(t) is the velocity function v(t); that is, the area under the curve of an acceleration vs. time (a vs. t) graph corresponds to the change of velocity.
Likewise, the integral of the jerk function j(t), the derivative of the acceleration function, can be used to find the change of acceleration at a certain time:
UnitsEdit
Acceleration has the dimensions of velocity (L/T) divided by time, i.e. L T−2. The SI unit of acceleration is the metre per second squared (m s−2); or "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.
Other formsEdit
An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, although its speed may be constant. In this case it is said to be undergoing centripetal (directed towards the center) acceleration.
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 F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large.