Acceleration

Last updated

Acceleration
Gravity gravita grave.gif
In vacuum (no air resistance), objects attracted by Earth gain speed at a steady rate.
Common symbols
a
SI unit m/s2, m·s−2, m s−2
Derivations from
other quantities
Dimension

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). [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:

Contents

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 orthogonal 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 and mathematically a negative, sometimes called deceleration, and passengers experience the reaction to deceleration as an inertial force pushing them forward. Such negative accelerations are often achieved by retrorocket burning in spacecraft. [4] Both acceleration and deceleration are treated the same, 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 vehicle. a=d\dt v

Definition and properties

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. Kinematics.svg
Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

Average acceleration

Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Dt - 0 of Dv/Dt Acceleration as derivative of velocity along trajectory.svg
Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0 of Δv/Δt

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

From bottom to top:
an acceleration function a(t);
the integral of the acceleration is the velocity function v(t);
and the integral of the velocity is the distance function s(t). 1-D kinematics.svg
From bottom to top:
  • an acceleration function a(t);
  • the integral of the acceleration is the velocity function v(t);
  • and the integral of the velocity is the distance function s(t).

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 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 velocity.

Likewise, the integral of the jerk function j(t), the derivative of the acceleration function, can be used to find acceleration at a certain time:

Units

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

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.

Tangential and centripetal acceleration

An oscillating pendulum, with velocity and acceleration marked. It experiences both tangential and centripetal acceleration. Oscillating pendulum.gif
An oscillating pendulum, with velocity and acceleration marked. It experiences both tangential and centripetal acceleration.
Components of acceleration for a curved motion. The tangential component at is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) ac is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path. Acceleration components.JPG
Components of acceleration for a curved motion. The tangential component at is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) ac is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.

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 ut, 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 un 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 three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas. [6] [7]

Special cases

Uniform acceleration

Calculation of the speed difference for a uniform acceleration Strecke und konstante Beschleunigung.png
Calculation of the speed difference for a 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 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 velocities, and acceleration to the time elapsed: [8]

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 a vacuum near the surface of Earth. [9]

Circular motion

Position vector plane polar coords.svg
Position vector r, always points radially from the origin.
Velocity vector plane polar coords.svg
Velocity vector v, always tangent to the path of motion.
Acceleration vector plane polar coords.svg
Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 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 to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to 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 to the center of the circle. This acceleration constantly changes the direction of the velocity to be tangent in the neighboring point, thereby rotating the velocity vector along the circle.

• For a given speed , the magnitude of this geometrically caused acceleration (centripetal acceleration) is inversely proportional to the radius of the circle, and increases as the square of this speed:

• Note that, for a given angular velocity , the centripetal acceleration is directly proportional to radius . This is due to the dependence of velocity on the radius .

Expressing centripetal acceleration vector in polar components, where is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yields

As usual in rotations, the speed of a particle may be expressed as an angular speed 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.

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 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 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. [10]

Conversions

Conversions between common units of acceleration
Base value(Gal, or cm/s2)(ft/s2)(m/s2)(Standard gravity, g0)
1 Gal, or cm/s210.03280840.010.00101972
1 ft/s230.480010.3048000.0310810
1 m/s21003.2808410.101972
1 g0980.66532.17409.806651

See also

Related Research Articles

A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

Jerk (physics) Rate of change of acceleration with time.

In physics, jerk or jolt is the rate at which an object's acceleration changes with respect to time. It is a vector quantity. Jerk is most commonly denoted by the symbol j and expressed in m/s3 or standard gravities per second (g0/s).

Torque Physics concept

In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study. The concept originated with the studies by Archimedes of the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M.

In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion of an object and the forces acting on it. The first law states that an object either remains at rest or continues to move at a constant velocity, unless it is acted upon by an external force. The second law states that the rate of change of momentum of an object is directly proportional to the force applied, or, for an object with constant mass, that the net force on an object is equal to the mass of that object multiplied by the acceleration. The third law states that when one object exerts a force on a second object, that second object exerts a force that is equal in magnitude and opposite in direction on the first object.

Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In physics, angular velocity, also known as angular frequency vector, is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.

In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceleration, called spin angular acceleration and orbital angular acceleration respectively. Spin angular acceleration refers to the angular acceleration of a rigid body about its centre of rotation, and orbital angular acceleration refers to the angular acceleration of a point particle about a fixed origin.

Work (physics) Process of energy transfer to an object via force application through displacement

In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.

Rigid body Physical object which does not deform when forces or moments are exerted on it

In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass.

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body. In circular motion, the distance between the body and a fixed point on the surface remains the same.

Fictitious force A force on objects moving within a reference frame that rotates with respect to an inertial frame.

A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as an accelerating or rotating reference frame. An example is seen in a passenger vehicle that is accelerating in the forward direction – passengers perceive that they are acted upon by a force in the rearward direction pushing them back into their seats. An example in a rotating reference frame is the force that appears to push objects outwards towards the rim of a centrifuge. These apparent forces are examples of fictitious forces.

Rotating reference frame

A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second2. In acoustics or physics, acceleration is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time2. In SI units, this is m/s2.

A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as .

Rotation around a fixed axis Type of motion

Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.

Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; and non-uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle along a line can be described by its position , which varies with (time). An example of linear motion is an athlete running 100m along a straight track.

Classical mechanics branch of physics concerned with the set of classical laws describing the non-relativistic motion of bodies under the action of a system of forces

Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility).

Isaac Newton's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical spheres. The basis of the argument is that all observers make two observations: the tension in the string joining the bodies and the rate of rotation of the spheres. Only for the truly non-rotating observer will the tension in the string be explained using only the observed rate of rotation. For all other observers a "correction" is required that accounts for the tension calculated being different from the one expected using the observed rate of rotation. It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space. Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to what?" General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.

In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

References

  1. Bondi, Hermann (1980). Relativity and Common Sense. Courier Dover Publications. pp.  3. ISBN   978-0-486-24021-3.
  2. Lehrman, Robert L. (1998). Physics the Easy Way. Barron's Educational Series. pp.  27. ISBN   978-0-7641-0236-3.
  3. Crew, Henry (2008). The Principles of Mechanics. BiblioBazaar, LLC. p. 43. ISBN   978-0-559-36871-4.
  4. Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2008). College Physics, Volume 10. Cengage. p. 32. ISBN   9780495386933.
  5. Weisstein, Eric W. "Chain Rule". Wolfram MathWorld. Wolfram Research. Retrieved 2 August 2016.
  6. Larry C. Andrews; Ronald L. Phillips (2003). Mathematical Techniques for Engineers and Scientists. SPIE Press. p. 164. ISBN   978-0-8194-4506-3.
  7. Ch V Ramana Murthy; NC Srinivas (2001). Applied Mathematics. New Delhi: S. Chand & Co. p. 337. ISBN   978-81-219-2082-7.
  8. Keith Johnson (2001). Physics for you: revised national curriculum edition for GCSE (4th ed.). Nelson Thornes. p. 135. ISBN   978-0-7487-6236-1.
  9. David C. Cassidy; Gerald James Holton; F. James Rutherford (2002). Understanding physics. Birkhäuser. p. 146. ISBN   978-0-387-98756-9.
  10. Brian Greene, The Fabric of the Cosmos: Space, Time, and the Texture of Reality , page 67. Vintage ISBN   0-375-72720-5