Time derivative

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 . A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation, A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E. (This is called Newton's notation) Higher time derivatives are also used: the second derivative with respect to time is written as with the corresponding shorthand of . As a generalization, the time derivative of a vector, say: is defined as the vector whose components are the derivatives of the components of the original vector. That is, Time derivatives are a key concept in physics. For example, for a changing position , its time derivative is its velocity, and its second derivative with respect to time, , is its acceleration. Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk. See motion graphs and derivatives. A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another: force is the time derivative of momentum power is the time derivative of energy electric current is the time derivative of electric charge and so on. A common occurrence in physics is the time derivative of a vector, such as velocity or displacement. In dealing with such a derivative, both magnitude and orientation may depend upon time. Uniform circular motion and Centripetal force For example, consider a particle moving in a circular path. Its position is given by the displacement vector , related to the angle, θ, and radial distance, r, as defined in the figure: For this example, we assume that θ = t. Hence, the displacement (position) at any time t is given by This form shows the motion described by r(t) is in a circle of radius r because the magnitude of r(t) is given by using the trigonometric identity sin2(t) + cos2(t) = 1 and where is the usual Euclidean dot product.