Jounce

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:

${\displaystyle \overrightarrow{s}=\frac{d\overrightarrow{ȷ}}{dt}=\frac{d^2\overrightarrow{a}}{d{t}^2}=\frac{d^3\overrightarrow{v}}{d{t}^3}=\frac{d^4\overrightarrow{r}}{d{t}^4}.}$

The following equations are used for constant jounce:

${\displaystyle \overrightarrow{ȷ}={\overrightarrow{ȷ}}_0+\overrightarrow{s}t,}$

${\displaystyle \overrightarrow{a}={\overrightarrow{a}}_0+{\overrightarrow{ȷ}}_{0}t+\frac{1}{2}\overrightarrow{s}{t}^2,}$

${\displaystyle \overrightarrow{v}={\overrightarrow{v}}_0+{\overrightarrow{a}}_{0}t+\frac{1}{2}{\overrightarrow{ȷ}}_0{t}^2+\frac{1}{6}\overrightarrow{s}{t}^3,}$

${\displaystyle \overrightarrow{r}={\overrightarrow{r}}_0+{\overrightarrow{v}}_{0}t+\frac{1}{2}{\overrightarrow{a}}_0{t}^2+\frac{1}{6}{\overrightarrow{ȷ}}_0{t}^3+\frac{1}{24}\overrightarrow{s}{t}^4,}$

where

${\displaystyle \overrightarrow{s}}$ is constant jounce,

${\displaystyle {\overrightarrow{ȷ}}_0}$ is initial jerk,

${\displaystyle \overrightarrow{ȷ}}$ is final jerk,

${\displaystyle {\overrightarrow{a}}_0}$ is initial acceleration,

${\displaystyle \overrightarrow{a}}$ is final acceleration,

${\displaystyle {\overrightarrow{v}}_0}$ is initial velocity,

${\displaystyle \overrightarrow{v}}$ is final velocity,

${\displaystyle {\overrightarrow{r}}_0}$ is initial position,

${\displaystyle \overrightarrow{r}}$ is final position,

${\displaystyle t}$ is time between initial and final states.

The notation ${\displaystyle \overrightarrow{s}}$ (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]

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

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