Acceleration

Acceleration, in physics, is the rate of change of velocity of an object. An object's acceleration is the net result of any and all forces acting on the object, as described by Newton's Second Law. The SI unit for acceleration is metre per second squared (m s^-2). Accelerations are vector quantities (they have magnitude and direction) and add according to the parallelogram law. As a vector, the calculated net force is equal to the product of the object's mass (a scalar quantity) and its acceleration.

For example, when a car starts from a standstill (zero relative velocity) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the car turns, there is an acceleration toward the new direction. In this example, we can call the forward acceleration of the car a "linear acceleration", which passengers in the car might experience as a force pushing them back into their seats. When changing direction, we might call this "non-linear acceleration", which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the opposite direction from the direction of the vehicle, sometimes called deceleration. Passengers may experience deceleration as a force lifting them forwards. Mathematically, there is no separate formula for deceleration: both are changes in velocity. Each of these accelerations (linear, non-linear, deceleration) might be felt by passengers until their velocity (speed and direction) matches that of the car.

Academic acceleration

Academic acceleration is the advancement of gifted students in subjects at a rate that places them ahead of where they would be in the regular school curriculum. Because it provides students with level-appropriate material, academic acceleration has been described as a "fundamental need" for gifted students. Although the bulk of educational research on academic acceleration has been within the United States, the practice occurs worldwide.

Impact

Well-administered academic acceleration programs have been generally found to be highly beneficial to students. Effective administration involves ensuring student readiness, both academic and emotional, and providing necessary support and resources. Cohort acceleration programs, in which a number of students are accelerated together at the same time, are often especially effective. However, acceleration programs often face difficulty due to many teachers, administrators and parents being skeptical of the benefits of acceleration. Adults who have experienced acceleration themselves, however, tend to be very well-disposed to the practice.

Acceleration (differential geometry)

In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend".

Formal definition

Consider a differentiable manifold with a given connection . Let be a curve in with tangent vector, i.e. velocity, , with parameter .

The acceleration vector of is defined by , where denotes the covariant derivative associated to .

It is a covariant derivative along , and it is often denoted by

With respect to an arbitrary coordinate system , and with being the components of the connection (i.e., covariant derivative ) relative to this coordinate system, defined by

for the acceleration vector field one gets:

where is the local expression for the path , and .

The concept of acceleration is a covariant derivative concept. In other words, in order to define acceleration an additional structure on must be given.

See also