Acceleration
Acceleration  

In the absence of air resistance, a falling ball would continue to accelerate.


Common symbols  a 
SI unit  m/s^{2}, m·s^{2}, m s^{2} 
Classical mechanics 


Fundamentals

Core topics

Acceleration, in physics, is the rate at which the velocity of an object changes over time.^{1} Velocity and acceleration are vector quantities, with magnitude and direction that add according to the parallelogram law.^{2}^{3} An object's acceleration, as described by Newton's Second Law, is due to the net force acting on the object, i.e., the net result of any and all forces acting on the object. As a vector, this net force is equal to the product of the object's mass (a scalar quantity) and the acceleration. The SI unit for acceleration is the metre per second squared (m/s^{2}).
For example, an object such as a car that starts from a standstill, then travels in a straight line at increasing speed, is accelerating in the direction of travel. If the car changes direction at constant speedometer reading, there is, strictly speaking, an acceleration although it is often not so described; passengers in the car will experience a force pushing them back into their seats in linear acceleration, and a sideways force on changing direction. If the speed of the car decreases, it is sometimes called deceleration; mathematically it is simply acceleration in the opposite direction to that of motion.^{4}
Contents
Definition and properties
Mathematically, instantaneous acceleration—acceleration over an infinitesimal interval of time—is the rate of change of velocity over time:
 i.e., the derivative of the velocity vector as a function of time.
(Here and elsewhere, if motion is in a straight line, vector quantities can be substituted by scalars in the equations.)
Average acceleration over a period of time is the change in velocity divided by the duration of the period
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}); this can be called more meaningfully "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.
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 the magnitude (speed) may be constant. When an object is executing such a motion where it changes direction, but not speed, it is said to be undergoing centripetal (directed towards the center) acceleration. Oppositely, a change in the speed of an object, but not its direction of motion, is a tangential acceleration.
Proper acceleration, the acceleration of a body relative to a freefall 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 centerofmass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large and acceleration becomes less.
Tangential and centripetal acceleration
The velocity of a particle moving on a curved path as a function of time can be written as:
with v(t) equal to the speed of travel along the path, and
a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed v(t) and the changing direction of u_{t}, the acceleration of a particle moving on a curved path can be written using the chain rule of differentiation^{5} for the product of two functions of time as:
where u_{n} is the unit (inward) normal vector to the particle's trajectory (also called the principal normal), and r is its instantaneous radius of curvature based upon the osculating circle at time t. These components are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also circular motion and centripetal force).
Geometrical analysis of threedimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas.^{6}^{7}
Special cases
Uniform acceleration
Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an equal amount in every equal time period.
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 g (also called acceleration due to gravity). By Newton's Second Law the force, F, acting on a body is given by:
Due to the simple algebraic properties of constant acceleration in the onedimensional case (that is, the case of acceleration aligned with the initial velocity), there are simple formulas relating the quantities displacement s, initial velocity v_{0}, final velocity v, acceleration a, and time t:^{8}
where
 = displacement
 = initial velocity
 = final velocity
 = uniform acceleration
 = time.
In the case of uniform acceleration of an object that is initially moving in a direction not aligned with the acceleration, 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, as in the trajectory of a cannonball, neglecting air resistance.^{9}
Circular motion
Uniform circular motion, that is constant speed along a circular path, is an example of a body experiencing acceleration resulting in velocity of a constant magnitude but change of direction. In this case, because the direction of the object's motion is constantly changing, being tangential to the circle, the object's linear velocity vector also changes, but its speed does not. This acceleration is a radial acceleration since it is always directed toward the centre of the circle and takes the magnitude:
where is the object's linear speed along the circular path. Equivalently, the radial acceleration vector () may be calculated from the object's angular velocity , whence:
where is a vector directed from the centre of the circle and equal in magnitude to the radius. The negative shows that the acceleration vector is directed towards the centre of the circle (opposite to the radius).
The acceleration, hence also the net force acting on a body in uniform circular motion, is directed toward the centre of the circle; that is, it is centripetal. Whereas the socalled 'centrifugal force' appearing to act outward on the body is really a pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum at a tangent to the circle.
With nonuniform circular motion, i.e., the speed along the curved path changes, a transverse accleration is produced equal to the rate of change of the angular speed around the circle times the radius of the circle. That is,
The transverse (or tangential) acceleration is directed at right angles to the radius vector and takes the sign of the angular acceleration ().
Relation to relativity
Special relativity
The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in a vacuum. Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations.
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.
General relativity
Unless the state of motion of an object is known, it is totally impossible to distinguish whether an observed force is due to gravity or to acceleration—gravity and inertial acceleration have identical effects. Albert Einstein called this the principle of equivalence, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating.^{10}
Conversions
m/s^{2}  ft/s^{2}  standard gravity (g_{0})  Gal (cm/s^{2})  

1 m/s^{2}  1  3.28084  0.101972  100 
1 ft/s^{2}  0.304800  1  0.0310810  30.4800 
g_{0}  9.80665  32.1740  1  980.665 
1 cm/s^{2}  0.01  0.0328084  0.00101972  1 
See also
 0 to 60 mph (0 to 100 km/h)
 Fourvector: making the connection between space and time explicit
 Gravitational acceleration
 Shock (mechanics)
 Shock and vibration data logger
measuring 3axis acceleration  Specific force
References
 ^ Crew, Henry (2008). The Principles of Mechanics. BiblioBazaar, LLC. p. 43. ISBN 0559368712.
 ^ Bondi, Hermann (1980). Relativity and Common Sense. Courier Dover Publications. p. 3. ISBN 0486240215.
 ^ Lehrman, Robert L. (1998). Physics the Easy Way. Barron's Educational Series. p. 27. ISBN 0764102362.
 ^ Raymond A. Serway, Chris Vuille, Jerry S. Faughn (2008). College Physics, Volume 10. Cengage. p. 32. ISBN 9780495386933.
 ^ "Chain Rule".
 ^ Larry C. Andrews & Ronald L. Phillips (2003). Mathematical Techniques for Engineers and Scientists. SPIE Press. p. 164. ISBN 0819445061.
 ^ Ch V Ramana Murthy & NC Srinivas (2001). Applied Mathematics. New Delhi: S. Chand & Co. p. 337. ISBN 8121920825.
 ^ Keith Johnson (2001). Physics for you: revised national curriculum edition for GCSE (4th ed.). Nelson Thornes. p. 135. ISBN 9780748762361.
 ^ David C. Cassidy, Gerald James Holton, and F. James Rutherford (2002). Understanding physics. Birkhäuser. p. 146. ISBN 9780387987569.
 ^ Brian Greene, The Fabric of the Cosmos, page 67. Vintage ISBN 0375727205
External links
Wikimedia Commons has media related to Acceleration. 
 Acceleration Calculator Simple acceleration unit converter
 Measurespeed.com  Acceleration Calculator Based on starting & ending speed and time elapsed.

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