Acceleration | |
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In vacuum (no air resistance), objects attracted by Earth gain tốc độ at a steady rate. | |
Common symbols Bạn đang xem: công thức tính gia tốc | a |
SI unit | m/s^{2}, m·s^{−2}, m s^{−2} |
Derivations from | |
Dimension |
In mechanics, acceleration is the rate of change of the velocity of an object with respect lớn time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are vector quantities (in that they have magnitude and direction).^{[1]}^{[2]} 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,^{[3]} is the combined effect of two causes:
- the net balance of all external forces acting onto that object — magnitude is directly proportional lớn this net resulting force;
- that object's mass, depending on the materials out of which it is made — magnitude is inversely proportional lớn 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 lớn 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 lớn which the passengers experience as a centrifugal force. If the tốc độ 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^{[4]}^{[5]} or retardation, and passengers experience the reaction lớn deceleration as an inertial force pushing them forward. Such negative accelerations are often achieved by retrorocket burning in spacecraft.^{[6]} 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 lớn the acceleration due lớn change in tốc độ.
Definition and properties[edit]
Average acceleration[edit]
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[edit]
Instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. In the terms of calculus, instantaneous acceleration is the derivative of the velocity vector with respect lớn time:
As acceleration is defined as the derivative of velocity, v, with respect lớn time t and velocity is defined as the derivative of position, x, with respect lớn time, acceleration can be thought of as the second derivative of x with respect lớn 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 lớn the change of velocity.
Likewise, the integral of the jerk function j(t), the derivative of the acceleration function, can be used lớn find the change of acceleration at a certain time:
Units[edit]
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 forms[edit]
An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due lớn the change of direction of motion, although its tốc độ may be constant. In this case it is said lớn be undergoing centripetal (directed towards the center) acceleration.
Proper acceleration, the acceleration of a toàn thân relative lớn a free-fall condition, is measured by an instrument called an accelerometer.
In classical mechanics, for a toàn thân with constant mass, the (vector) acceleration of the body's center of mass is proportional lớn 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 toàn thân, m is the mass of the toàn thân, and a is the center-of-mass acceleration. As speeds approach the tốc độ of light, relativistic effects become increasingly large.
Tangential and centripetal acceleration[edit]
The velocity of a particle moving on a curved path as a function of time can be written as:
with v(t) equal lớn the tốc độ of travel along the path, and
a unit vector tangent lớn the path pointing in the direction of motion at the chosen moment in time. Taking into trương mục both the changing tốc độ 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^{[7]} for the product of two functions of time as:
where u_{n} is the unit (inward) normal vector lớn 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. The 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), respectively.
Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas.^{[8]}^{[9]}
Special cases[edit]
Uniform acceleration[edit]
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 không tính phí fall in a uniform gravitational field. The acceleration of a falling toàn thân in the absence of resistances lớn motion is dependent only on the gravitational field strength g (also called acceleration due lớn gravity). By Newton's Second Law the force acting on a toàn thân is given by:
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Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating the displacement, initial and time-dependent velocities, and acceleration lớn the time elapsed:^{[10]}
where
In particular, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according lớn 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.^{[11]}
Circular motion[edit]
Position vector r, always points radially from the origin.
Velocity vector v, always tangent lớn the path of motion.
Acceleration vector a, not parallel lớn the radial motion but offset by the angular and Coriolis accelerations, nor tangent lớn the path but offset by the centripetal and radial accelerations.
Kinematic vectors in plane polar coordinates. Notice the setup is not restricted lớn 2d space, but may represent the osculating plane plane in a point of an arbitrary curve in any higher dimension.
In uniform circular motion, that is moving with constant speed along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant. The derivative of the location of a point on a curve with respect lớn time, i.e. its velocity, turns out lớn be always exactly tangential lớn the curve, respectively orthogonal lớn the radius in this point. Since in uniform motion the velocity in the tangential direction does not change, the acceleration must be in radial direction, pointing lớn the center of the circle. This acceleration constantly changes the direction of the velocity lớn be tangent in the neighboring point, thereby rotating the velocity vector along the circle.
Expressing centripetal acceleration vector in polar components, where is a vector from the centre of the circle lớn the particle with magnitude equal lớn this distance, and considering the orientation of the acceleration towards the center, yields
As usual in rotations, the tốc độ of a particle may be expressed as an angular speed with respect lớn 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 lớn keep it in this uniform circular motion. The so-called 'centrifugal force', appearing lớn act outward on the toàn thân, is a so-called pseudo force experienced in the frame of reference of the toàn thân in circular motion, due lớn the body's linear momentum, a vector tangent lớn the circle of motion.
In a nonuniform circular motion, i.e., the tốc độ along the curved path is changing, the acceleration has a non-zero component tangential lớn the curve, and is not confined lớn the principal normal, which directs lớn 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 tốc độ 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 lớn the radius vector.
Coordinate systems[edit]
In multi-dimensional Cartesian coordinate systems, acceleration is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding acceleration components are defined as^{[12]}
The two-dimensional acceleration vector is then defined as . The magnitude of this vector is found by the distance formula as
In three-dimensional systems where there is an additional z-axis, the corresponding acceleration component is defined as
The three-dimensional acceleration vector is defined as with its magnitude being determined by
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Relation lớn relativity[edit]
Special relativity[edit]
The special theory of relativity describes the behavior of objects traveling relative lớn other objects at speeds approaching that of light in vacuum. Newtonian mechanics is exactly revealed lớn be an approximation lớn reality, valid lớn great accuracy at lower speeds. As the relevant speeds increase toward the tốc độ 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 tốc độ is approached; an object with mass can approach this tốc độ asymptotically, but never reach it.
General relativity[edit]
Unless the state of motion of an object is known, it is impossible lớn distinguish whether an observed force is due lớn gravity or lớn acceleration—gravity and inertial acceleration have identical effects. Albert Einstein called this the equivalence principle, 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.^{[13]}
Conversions[edit]
Base value | (Gal, or cm/s^{2}) | (ft/s^{2}) | (m/s^{2}) | (Standard gravity, g_{0}) |
---|---|---|---|---|
1 Gal, or cm/s^{2} | 1 | 0.0328084 | 0.01 | 1.01972×10^{−3} |
1 ft/s^{2} | 30.4800 | 1 | 0.304800 | 0.0310810 |
1 m/s^{2} | 100 | 3.28084 | 1 | 0.101972 |
1 g_{0} | 980.665 | 32.1740 | 9.80665 | 1 |
See also[edit]
References[edit]
- ^ Bondi, Hermann (1980). Relativity and Common Sense. Courier Dover Publications. pp. 3. ISBN 978-0-486-24021-3.
- ^ Lehrman, Robert L. (1998). Physics the Easy Way. Barron's Educational Series. pp. 27. ISBN 978-0-7641-0236-3.
- ^ Crew, Henry (2008). The Principles of Mechanics. BiblioBazaar, LLC. p. 43. ISBN 978-0-559-36871-4.
- ^ P. Smith; R. C. Smith (1991). Mechanics (2nd, illustrated, reprinted ed.). John Wiley & Sons. p. 39. ISBN 978-0-471-92737-2. Extract of page 39
- ^ John D. Cutnell; Kenneth W. Johnson (2014). Physics, Volume One: Chapters 1-17, Volume 1 (1st0, illustrated ed.). John Wiley & Sons. p. 36. ISBN 978-1-118-83688-0. Extract of page 36
- ^ Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2008). College Physics, Volume 10. Cengage. p. 32. ISBN 9780495386933.
- ^ Weisstein, Eric W. "Chain Rule". Wolfram MathWorld. Wolfram Research. Retrieved 2 August 2016.
- ^ Larry C. Andrews; Ronald L. Phillips (2003). Mathematical Techniques for Engineers and Scientists. SPIE Press. p. 164. ISBN 978-0-8194-4506-3.
- ^ Ch V Ramana Murthy; NC Srinivas (2001). Applied Mathematics. New Delhi: S. Chand & Co. p. 337. ISBN 978-81-219-2082-7.
- ^ Keith Johnson (2001). Physics for you: revised national curriculum edition for GCSE (4th ed.). Nelson Thornes. p. 135. ISBN 978-0-7487-6236-1.
- ^ David C. Cassidy; Gerald James Holton; F. James Rutherford (2002). Understanding physics. Birkhäuser. p. 146. ISBN 978-0-387-98756-9.
- ^ "The Feynman Lectures on Physics Vol. I Ch. 9: Newton's Laws of Dynamics". www.feynmanlectures.caltech.edu. Retrieved 2024-01-04.
- ^ Brian Greene, The Fabric of the Cosmos: Space, Time, and the Texture of Reality, page 67. Vintage ISBN 0-375-72720-5
External links[edit]
- Acceleration Calculator Simple acceleration unit converter
- Acceleration Calculator Acceleration Conversion calculator converts units sườn meter per second square, kilometer per second square, millimeter per second square & more with metric conversion.
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