# Pop (physics)

Integrals and derivatives of displacement, including pop, as well as integrals and derivatives of energy, including actergy.[1]

In physics, pop, also known as pounce, is the sixth derivative of the position vector with respect to time, with the first, second, third, fourth, and fifth derivatives being velocity, acceleration, jerk, snap, and crackle, respectively; pop is thus the rate of change of the crackle with respect to time.[2][3] Pop is defined by any of the following equivalent expressions:

${\displaystyle {\vec {p}}={\frac {d{\vec {c}}}{dt}}={\frac {d^{2}{\vec {s}}}{dt^{2}}}={\frac {d^{3}{\vec {\jmath }}}{dt^{3}}}={\frac {d^{4}{\vec {a}}}{dt^{4}}}={\frac {d^{5}{\vec {v}}}{dt^{5}}}={\frac {d^{6}{\vec {r}}}{dt^{6}}}}$

The following equations are used for constant pop:

${\displaystyle {\vec {c}}={\vec {c}}_{0}+{\vec {p}}\,t}$
${\displaystyle {\vec {s}}={\vec {s}}_{0}+{\vec {c}}_{0}\,t+{\frac {1}{2}}{\vec {p}}\,t^{2}}$
${\displaystyle {\vec {\jmath }}={\vec {\jmath }}_{0}+{\vec {s}}_{0}\,t+{\frac {1}{2}}{\vec {c}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {p}}\,t^{3}}$
${\displaystyle {\vec {a}}={\vec {a}}_{0}+{\vec {\jmath }}_{0}\,t+{\frac {1}{2}}{\vec {s}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {c}}_{0}\,t^{3}+{\frac {1}{24}}{\vec {p}}\,t^{4}}$
${\displaystyle {\vec {v}}={\vec {v}}_{0}+{\vec {a}}_{0}\,t+{\frac {1}{2}}{\vec {\jmath }}_{0}\,t^{2}+{\frac {1}{6}}{\vec {s}}_{0}\,t^{3}+{\frac {1}{24}}{\vec {c}}_{0}\,t^{4}+{\frac {1}{120}}{\vec {p}}\,t^{5}}$
${\displaystyle {\vec {r}}={\vec {r}}_{0}+{\vec {v}}_{0}\,t+{\frac {1}{2}}{\vec {a}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {\jmath }}_{0}\,t^{3}+{\frac {1}{24}}{\vec {s}}_{0}\,t^{4}+{\frac {1}{120}}{\vec {c}}_{0}\,t^{5}+{\frac {1}{720}}{\vec {p}}\,t^{6}}$

where

${\displaystyle {\vec {p}}}$ : constant pop,
${\displaystyle {\vec {c}}_{0}}$ : initial crackle,
${\displaystyle {\vec {c}}}$ : final crackle,
${\displaystyle {\vec {s}}_{0}}$ : initial snap,
${\displaystyle {\vec {s}}}$ : final snap,
${\displaystyle {\vec {\jmath }}_{0}}$ : initial jerk,
${\displaystyle {\vec {\jmath }}}$ : final jerk,
${\displaystyle {\vec {a}}_{0}}$ : initial acceleration,
${\displaystyle {\vec {a}}}$ : final acceleration,
${\displaystyle {\vec {v}}_{0}}$ : initial velocity,
${\displaystyle {\vec {v}}}$ : final velocity,
${\displaystyle {\vec {r}}_{0}}$ : initial position,
${\displaystyle {\vec {r}}}$ : final position,
${\displaystyle t}$ : time between initial and final states.

The terms snap (also referred to as jounce), crackle, and pop‍—‌for the fourth, fifth, and sixth derivatives of position‍—‌were inspired by the advertising mascots Snap, Crackle, and Pop.[3]

## Unit and dimension

The dimensions of pop are LT−6. In SI units, this is m/s6, and in CGS units, 100 Gal per quartic second. This pattern[clarification needed] continues for higher order derivatives.

## References

1. ^ Janzen, Ryan; Mann, Steve; et al. (2014). "Actergy as a Reflex Performance Metric: Integral-Kinematics Applications". Proceedings of the IEEE GEM 2014: 311–2. doi:10.1109/GEM.2014.7048123.
2. ^ Thompson, Peter M. (5 May 2011). "Snap, Crackle, and Pop" (PDF). AIAA Info. Hawthorne, California: Systems Technology. p. 1. Retrieved 3 March 2017. The common names for the first three derivatives are velocity, acceleration, and jerk. The not so common names for the next three derivatives are snap, crackle, and pop.
3. ^ a b Visser, Matt (31 March 2004). "Jerk, snap and the cosmological equation of state" (PDF). Classical and Quantum Gravity. 21 (11): 2603–2616. arXiv:gr-qc/0309109. Bibcode:2004CQGra..21.2603V. doi:10.1088/0264-9381/21/11/006. ISSN 0264-9381. Retrieved 17 May 2015. Snap [the fourth time derivative] is also sometimes called jounce. The fifth and sixth time derivatives are sometimes somewhat facetiously referred to as crackle and pop.