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Concept
Time derivative
Summary
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.
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