Jounce

In physics, jounce or snap is the fourth derivative of the position vector with respect to time, with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively; in other words, the jounce is the rate of change of the jerk with respect to time. Jounce is defined by any of the following equivalent expressions:

$\vec s =\frac {d \vec j} {dt}=\frac {d^2 \vec a} {dt^2}=\frac {d^3 \vec v} {dt^3}=\frac {d^4 \vec r} {dt^4}$

The following equations are used for constant jounce:

$\vec j = \vec j_0 + \vec s \,t$
$\vec a = \vec a_0 + \vec j_0 \,t + \frac{1}{2} \vec s \,t^2$
$\vec v = \vec v_0 + \vec a_0 \,t + \frac{1}{2} \vec j_0 \,t^2 + \frac{1}{6} \vec s \,t^3$
$\vec r = \vec r_0 + \vec v_0 \,t + \frac{1}{2} \vec a_0 \,t^2 + \frac{1}{6} \vec j_0 \,t^3 + \frac{1}{24} \vec s \,t^4$

where

$\vec s$ : constant jounce,
$\vec j_0$ : initial jerk,
$\vec j$ : final jerk,
$\vec a_0$ : initial acceleration,
$\vec a$ : final acceleration,
$\vec v_0$ : initial velocity,
$\vec v$ : final velocity,
$\vec r_0$ : initial position,
$\vec r$ : final position,
$t$ : time between initial and final states.

The notation $\vec s$ (used in [1]) is not to be confused with the displacement vector commonly denoted similarly. Currently, there are no well-accepted designations for the derivatives of jounce. The fourth, fifth and sixth derivatives of position as a function of time are "sometimes somewhat facetiously"[1][2] referred to as "Snap", "Crackle", and "Pop" respectively. Because higher-order derivatives are not commonly useful, there has been no consensus among physicists on the proper names for derivatives above jounce.[2] Despite this, physicists have proposed other names such as "Lock", "Drop", "Shot" and "Put" for higher-ordered derivatives.

The dimensions of jounce are distance per (time to the power of 4). In SI units, this is "metres per quartic second", "metres per second per second per second per second", m/s4, m · s−4, or 100 Gal per second squared in CGS units. This pattern continues for higher order derivatives, with the 5th being m/s5. One arbitrary example is the 10th derivative. Although technically correct, the dimensions for the tenth derivative could be somewhat ridiculously defined as "metres per second per second per second per second per second per second per second per second per second per second" or more simply m/s10.

References

1. ^ a b Visser, Matt (2004-07-24). "Jerk, Snap, and the Cosmological Equation of State". Classical and Quantum Gravity 21 (11): 2603–2616. arXiv:gr-qc/0309109. Bibcode:2004CQGra..21.2603V. doi:10.1088/0264-9381/21/11/006.
2. ^ a b Gragert, Stephanie (November 1998). "What is the term used for the third derivative of position?". Usenet Physics and Relativity FAQ. Math Dept., University of California, Riverside. Retrieved 2008-03-12.