Pop (physics)

In physics, pop 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. ^{[1] [2]} 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. ^[2]

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.

References

1. Thompson, Peter M. (March 2011). "Snap, Crackle, and Pop" (PDF). Proc of AIAA Southern California Aerospace Systems and Technology Conference. p. 1. Archived from the original (PDF) on 2017-03-04. Retrieved 29 February 2020. 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.
2. 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.