Acceleration

"Accelerate" redirects here. For other uses, see Accelerate (disambiguation).

In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. For example, a car driving away (from standstill) is increasing its speed and is thus accelerating. Similarly, a car braking to stop in front of a traffic light is still said (in physics) to undergo acceleration, although now a negative one. In common speech, it is said to be decelerating.

However, velocity has both a magnitude and direction (i.e. it is a vector), and thus acceleration is also a vector. As such, it describes the rate of change of both the magnitude (the speed) and the direction of velocity. This means that an object moving in a circular motion—such as a satellite orbiting the earth—is also accelerating, even though it may be moving at a constant speed. 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.

Acceleration has the dimensions L T ⁻². In SI units, acceleration is measured in meters per second squared (m/s²).

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 acceleration of the body is proportional to the net force acting on it (Newton's second law):

$\mathbf{F} = m\mathbf{a} \quad \to \quad \mathbf{a} = \mathbf{F}/m$

where F is the resultant force acting on the body, m is the mass of the body, and a is its 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 Δt → 0.

Components of acceleration for a planar curved motion. The tangential component a_t is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector. The centripetal component a_c 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.

Average acceleration is the change in velocity (Δv) divided by the change in time (Δt). Instantaneous acceleration is the acceleration at a specific point in time which is for a very short interval of time as Δt approaches zero. Acceleration can therefore be computed as the derivative (with respect to time) of velocity.

Tangential and centripetal acceleration

See also: Local coordinates

The velocity of a particle moving on a curved path as a function of time can be written as:

$\mathbf{v} (t) =v(t) \frac {\mathbf{v}(t)}{v(t)} = v(t) \mathbf{u}_\mathrm{t}(t) ,$

with v(t) equal to the speed of travel along the path, and

$\mathbf{u}_\mathrm{t} = \frac {\mathbf{v}(t)}{v(t)} \ ,$

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 u_t, 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:

$\begin{alignat}{3} \mathbf{a} & = \frac{d \mathbf{v}}{dt} \\ & = \frac{\mathrm{d}v }{\mathrm{d}t} \mathbf{u}_\mathrm{t} +v(t)\frac{d \mathbf{u}_\mathrm{t}}{dt} \\ & = \frac{\mathrm{d}v }{\mathrm{d}t} \mathbf{u}_\mathrm{t}+ \frac{v^2}{R}\mathbf{u}_\mathrm{n}\ , \\ \end{alignat}$

where u_n 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 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:

$\mathbf {F} = m \mathbf {g}$

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 formulas that relate the following quantities: displacement, initial velocity, final velocity, acceleration, and time:

$\mathbf {v}= \mathbf {u} + \mathbf {a} t$

$\mathbf {s}= \mathbf {u} t+ {{1} \over {2}} \mathbf {a}t^2 = {{(\mathbf{u}+\mathbf{v})t} \over {2}}$

$|\mathbf {v}|^2= |\mathbf {u}|^2 + 2 \, \mathbf {a} \cdot \mathbf {s}$

where

$\mathbf{s}$ = displacement

$\mathbf{u}$ = initial velocity

$\mathbf{v}$ = final velocity

$\mathbf{a}$ = 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

Uniform circular motion 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 velocity also changes, but its speed does not. This acceleration is directed toward the centre of the circle and takes the value:

$a = {{v^2} \over {r}}$

where v is the object's speed. Equivalently, the radial acceleration may be calculated from the object's angular velocity $\omega$, whence:

$\mathbf {a}= {-\omega^2} \mathbf {r}.$

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

"The force one feels from gravity and the force one feels from acceleration are the same. They are equivalent. Einstein called this the principle of equivalence. Since gravity and acceleration are equivalent, if you feel gravity's influence, you must be accelerating. Einstein argued that only those observers who feel no force at all - including the force of gravity - are justified in declaring that they are not accelerating."

See also

• Angular acceleration
• Gravitational acceleration
• Kinematics
• Equations of motion
• Proper acceleration
• 0 to 60 mph (0 to 100 km/h)
• Shock (mechanics)
• Shock and vibration data logger measuring 3-axis acceleration
• Specific force

External links

• Acceleration Calculator Simple acceleration unit converter
• Measurespeed.com - Acceleration Calculator Based on starting & ending speed and time elapsed.