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Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
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 .
As =, the first time derivatives of inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide. Carrying out the differentiations and re-arranging some terms yields the acceleration relative to the rotating reference frame, a r {\displaystyle \mathbf {a} _{\mathrm {r} }}
Absement changes as an object remains displaced and stays constant as the object resides at the initial position. It is the first time-integral of the displacement [3] [4] (i.e. absement is the area under a displacement vs. time graph), so the displacement is the rate of change (first time-derivative) of the absement.
Further time derivatives have also been named, as snap or jounce (fourth derivative), crackle (fifth derivative), and pop (sixth derivative). [12] [13] The seventh derivative is known as "Bang," as it is a logical continuation to the cycle. The eighth derivative has been referred to as "Boom," and the 9th is known as "Crash."
Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
In the limit that the time interval approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, = = = ^ + ^ + ^. Thus, a particle's velocity is the time rate of change of its position.
An example of a generalized coordinate would be to describe the position of a pendulum using the angle of the pendulum relative to vertical, rather than by the x and y position of the pendulum. Although there may be many possible choices for generalized coordinates for a physical system, they are generally selected to simplify calculations ...