# Jounce

In physics, jounce or snap is the fourth derivative of the position vector with respect to time or the rate of change of the jerk with respect to time. Equivalently, it is second derivative of acceleration or the third derivative of velocity. Jounce is defined by any of the following equivalent expressions:

${\displaystyle {\vec {s}}={\frac {d\,{\vec {\jmath }}}{dt}}={\frac {d^{2}{\vec {a}}}{dt^{2}}}={\frac {d^{3}{\vec {v}}}{dt^{3}}}={\frac {d^{4}{\vec {r}}}{dt^{4}}}.}$

The following equations are used for constant jounce:

${\displaystyle {\vec {\jmath }}={\vec {\jmath }}_{0}+{\vec {s}}t,}$
${\displaystyle {\vec {a}}={\vec {a}}_{0}+{\vec {\jmath }}_{0}t+{\frac {1}{2}}{\vec {s}}t^{2},}$
${\displaystyle {\vec {v}}={\vec {v}}_{0}+{\vec {a}}_{0}t+{\frac {1}{2}}{\vec {\jmath }}_{0}t^{2}+{\frac {1}{6}}{\vec {s}}t^{3},}$
${\displaystyle {\vec {r}}={\vec {r}}_{0}+{\vec {v}}_{0}t+{\frac {1}{2}}{\vec {a}}_{0}t^{2}+{\frac {1}{6}}{\vec {\jmath }}_{0}t^{3}+{\frac {1}{24}}{\vec {s}}t^{4},}$

where

${\displaystyle {\vec {s}}}$ is constant jounce,
${\displaystyle {\vec {\jmath }}_{0}}$ is initial jerk,
${\displaystyle {\vec {\jmath }}}$ is final jerk,
${\displaystyle {\vec {a}}_{0}}$ is initial acceleration,
${\displaystyle {\vec {a}}}$ is final acceleration,
${\displaystyle {\vec {v}}_{0}}$ is initial velocity,
${\displaystyle {\vec {v}}}$ is final velocity,
${\displaystyle {\vec {r}}_{0}}$ is initial position,
${\displaystyle {\vec {r}}}$ is final position,
${\displaystyle t}$ is time between initial and final states.

The notation ${\displaystyle {\vec {s}}}$ (used by Visser[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]

The dimensions of jounce are distance per fourth power of time. In SI units, this is "metres per quartic 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.

## 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 2015-10-24.