Pop (physics)

In physics, pop is the sixth derivative of the position vector with respect to time, with the first, second, and third, fourth, and fifth derivatives being velocity, acceleration, jerk, snap (or jounce), and crackle, respectively; in other words, the pop is 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:

${\vec {c}}={\vec {c}}_{0}+{\vec {p}}\,t$

${\vec {s}}={\vec {s}}_{0}+{\vec {c}}_{0}\,t+{\frac {1}{2}}{\vec {p}}\,t^{2}$

${\displaystyle {\vec {\jmath }}={\vec {\jmath }}_{0}+{\vec {s}}_{0}\,t+{\frac {1}{2}}{\vec {c}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {p}}\,t^{3}}$

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

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

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

where

${\vec {p}}$ : constant pop,

${\vec {c}}_{0}$ : initial crackle,

${\vec {c}}$ : final crackle,

${\vec {s}}_{0}$ : initial jounce,

${\vec {s}}$ : final jounce,

${\displaystyle {\vec {\jmath }}_{0}}$ : initial jerk,

${\displaystyle {\vec {\jmath }}}$ : 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 name "pop", along with "snap" (also referred to as jounce) and "crackle" are somewhat facetious terms for the fourth, fifth, and sixth derivatives of position, being a reference to Snap, Crackle, and Pop. Currently, there are no well-accepted designations for the derivatives of pop. Higher-order derivatives of position are not commonly useful. Thus, there has been no consensus among physicists on the proper names for derivatives above pop. Despite this, physicists have proposed other names such as "lock", "drop", "shot", and "put" for seventh, eighth, ninth and tenth derivatives. The eleventh derivative is frequently referred to as the "Clint" . The twelfth derivative is uncommonly referred to as the "Roy" .

Unit and dimension

The dimension of pop LT⁻⁶. In SI units, this is "metres per hexic second", "metres per second per second per second per second per second per second", m/s⁶, m · s⁻⁶, or 100 Gal per quartric second in CGS units. This pattern continues for higher order derivatives.

References

1. Thompson, Peter M. (5 May 2011). "Snap, Crackle, and Pop" (PDF). AIAA Info. Hawthorne, California: Systems Technology. p. 1. Retrieved 17 May 2015. 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. Victoria University of Wellington. 21 (11): 4. 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.