# Fourth, fifth, and sixth derivatives of position

Worked example: motion problems (with definite integrals) | AP Calculus AB | Khan Academy
Worked example: motion problems (with definite integrals) | AP Calculus AB | Khan Academy

The fourth derivative is often referred to as snap or jounce. The name “snap” for the fourth derivative led to crackle and pop for the fifth and sixth derivatives respectively,[4] inspired by the Rice Krispies mascots Snap, Crackle, and Pop.[5] These terms are occasionally used, though “sometimes somewhat facetiously”.[5]

In civil engineering, the design of railway tracks and roads involves the minimization of snap, particularly around bends with different radii of curvature. When snap is constant, the jerk changes linearly, allowing for a smooth increase in radial acceleration, and when, as is preferred, the snap is zero, the change in radial acceleration is linear. The minimization or elimination of snap is commonly done using a mathematical clothoid function. Minimizing snap improves the performance of machine tools and roller coasters.[1]

The following equations are used for constant snap:

where

is constant snap,

is initial jerk,

is final jerk,

is initial acceleration,

is final acceleration,

is initial velocity,

is final velocity,

is initial position,

is final position,

is time between initial and final states.

The notation (used by Visser[5]) is not to be confused with the displacement vector commonly denoted similarly.

The dimensions of snap 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.

The fifth derivative of the position vector with respect to time is sometimes referred to as crackle.[4] It is the rate of change of snap with respect to time.[4][5] Crackle is defined by any of the following equivalent expressions:

The following equations are used for constant crackle:

where

: constant crackle,

: initial snap,

: final snap,

: initial jerk,

: final jerk,

: initial acceleration,

: final acceleration,

: initial velocity,

: final velocity,

: initial position,

: final position,

: time between initial and final states.

The dimensions of crackle are LT−5. In SI units, this is m/s5, and in CGS units, 100 gal per cubed second.

The sixth derivative of the position vector with respect to time is sometimes referred to as pop.[4] It is the rate of change of crackle with respect to time.[4][5] Pop is defined by any of the following equivalent expressions:

The following equations are used for constant pop:

where

: constant pop,

: initial crackle,

: final crackle,

: initial snap,

: final snap,

: initial jerk,

: final jerk,

: initial acceleration,

: final acceleration,

: initial velocity,

: final velocity,

: initial position,

: final position,

: time between initial and final states.

The dimensions of pop are LT−6. In SI units, this is m/s6, and in CGS units, 100 gal per quartic second.

^ abcdefThompson, 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.{{cite web}}: CS1 maint: unfit URL (link)

You are watching: Fourth, fifth, and sixth derivatives of position. Info created by THVinhTuy selection and synthesis along with other related topics.

Rate this post