Jounce

Integrals and derivatives of displacement, including jounce, as well as integrals and derivatives of energy, including actergy. (Janzen et al. 2014)

In physics, jounce, also known as 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. ^{[1] [2]} Equivalently, it is the 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 ^[2] ) is not to be confused with the displacement vector commonly denoted similarly.

The dimensions of jounce are distance per fourth power of time. In SI units, this is "metres per second to the fourth", m/s⁴, m⋅s⁻⁴, or 100 gal per second squared in CGS units.

Jounce and the fifth and sixth derivatives of position as a function of time are "sometimes somewhat facetiously" ^{[2] [3]} referred to as snap, crackle, and pop respectively. However, time derivatives of position of higher order than four appear rarely. ^[3]

References

1. Thompson, Peter M. (5 May 2011). "Snap, Crackle, and Pop" (PDF). AIAA Info. Hawthorne, California: Systems Technology. p. 1. Archived from the original on 26 June 2018. 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.
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.
3. 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.

External links

Look up jounce in Wiktionary, the free dictionary.

• Cosmography: cosmology without the Einstein equations, Matt Visser, School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, 2004.