Acceleration

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The rate of change of velocity is known as Acceleration.

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Acceleration The rate of change of velocity is known as acceleration. Acceleration, like velocity, is a vector quantity: it has both magnitude and direction. The velocity of an object moving on a straight path can change in magnitude only, so its acceleration is the rate of change of its speed. On a curved path, the velocity may or may not change in magnitude, but it will always change in direction, which means that the acceleration of an object moving on a curved path can never be zero. If velocity is stated in metres per second (m/s) and the time interval in seconds (s), then the units of acceleration are metres per second per second (m/s/s, or m/s2). The time rate of change of velocity. Since velocity is a directed or vector quantity involving both magnitude and direction, a velocity may change by a change of magnitude (speed) or by a change of direction or both. It follows that acceleration is also a directed, or vector, quantity. If the magnitude of the velocity of a body changes from VI ft/s to ft/s in t seconds, then the average acceleration a has a magnitude given by Eq. (1): a 1. velocity change elapsed time h At' At To designate it fully the direction should be given, as well as the magnitude. Instantaneous acceleration is defined as the limit of the ratio of the velocity change to the elapsed time as the time interval approaches zero. When the acceleration is constant, the average acceleration and the instantaneous acceleration are equal. Whenever a body is acted upon by an unbalanced force, it will undergo acceleration. If it is moving in a constant direction, the acting force will produce a continuous change in speed. If it is moving with a constant speed, the acting force will produce an acceleration consisting of a continuous change of direction. In the general case, the acting force may produce both a change of speed and a change of direction. Angular acceleration is a vector quantity representing the rate of change of angular velocity of a body experiencing rotational motion. If, for example, at an instant ti, a rigid body is rotating about an axis with an angular velocity 01, and at a later time h, it has an angular velocity 02, the average angular acceleration u is given by Eq. (2), in radians per second per second. 2. At The instantaneous angular acceleration is given by u do/dt.
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When a body moves in a circular path with constant linear speed at each point in its path, it is also being constantly accelerated toward the center of the circle under the action of the force required to constrain it to move in its circular path. This acceleration toward the center of path is called radial acceleration. The component of linear acceleration tangent to the path of a particle subject to an angular acceleration about the axis of rotation is called tangential acceleration. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. Acceleration has the dimensions L T 2. In SI units, acceleration is measured in meters per second squared (m/s2). (Negative acceleration i.e. retardation, also has the same dimensions/units.) Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an accelerometer. In common speech, the term acceleration is used for an increase in speed (the magnitude of velocity); a decrease in speed is called deceleration. In physics, a change in the direction of velocity also is acceleration: for rotary motion, the change in direction of velocity results in centripetal (toward the center) acceleration; whereas the rate of change of speed is a tangential acceleration. In classical mechanics, for a body with constant mass, the acceleration of the body is proportional to the net force acting on it (Newton's second law): F = ma a = F/m where F is the resultant force acting on the body, m is the mass of the body, and a is its acceleration. Average and instantaneous acceleration Average acceleration is the change in velocity (Av) divided by the change in time (At). Instantaneous acceleration is the acceleration at a specific point in time which is for a very short interval of time as At approaches zero. Tangential and centripetal acceleration The velocity of a particle moving on a curved path as a function of time can be written as:
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v(t) with v(t) equal to the speed of travel along the path, and v(t) 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 on a planar surface can be written using the chain rule of differentiation and the derivative of the product of two functions of time as: dv a dt dv dut ut + v(t) dt dt 2 du dt where un is the unit (inward) normal vector to the particle's trajectory, 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 radial acceleration or centripetal acceleration (see also circular motion and centripetal force). Extension of this approach to three-dimensional space curves that cannot be contained on a planar surface leads to the Frenet - Serret formulas. 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
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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: F — mg Due to the simple algebraic properties of constant acceleration in the one-dimensional case (that is, the case of acceleration aligned with the initial velocity), there are simple formulae that relate the following quantities: displacement, initial velocity, final velocity, acceleration, and time: v = 'u + at 1 2 s ut + —at 2 s = displacement u = initial velocity V = final velocity (u + v)t 2 •s where a = uniform acceleration t = 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. Circular motion An example of a body experiencing acceleration of a uniform magnitude but changing direction is uniform circular motion. In this case, because the direction of the object's motion is constantly changing, being tangential to the circle, the object's velocity also changes, but its speed does not. This acceleration is directed toward the centre of the circle and takes the value: 2 a
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where v is the object's speed. Equivalently, the radial acceleration may be calculated from the object's angular velocity o, whence: 2 The acceleration, hence also the force acting on a body in uniform circular motion, is directed toward the center of the circle; that is, it is centripetal — the so called 'centrifugal force' appearing to act outward on a 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. Relation to relativity After completing his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant proper acceleration are actually feeling themselves being accelerated, so that, for example, a car's forward acceleration would result in the driver feeling a slight pressure between himself and his seat. In the case of gravity, which Einstein concluded is not actually a force, this is not the case; acceleration due to gravity is not felt by an object in free-fall. The reason for this difference is that in the case of the car the force due to the engine is applied directly only to a certain part of the mass while the driver and the bulk of the car are passively dragged along. Gravity on the other hand accelerates the entire mass, with no internal forces acting. This was the basis for his development of general relativity, a relativistic theory of gravity.

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