Updated by Stephanie Gragert, 1998.
Original by Philip Gibbs, 1996.

# What term is used for the third derivative of displacement?

The first derivative of displacement x with respect to time is velocity v, and the second derivative is acceleration a.  Less well known is that the third derivative of displacement (and so the rate of increase of acceleration), is technically known as jerk j.  Jerk is a vector, but may also be used loosely as a scalar quantity because there is no separate term for the magnitude of jerk analogous to speed for magnitude of velocity.

In the UK, jolt has sometimes been used instead of jerk, and is equally acceptable.

Many other terms have appeared in individual cases for the third derivative, including pulse, impulse, bounce, surge, shock and super acceleration.  These are generally less appropriate than jerk and jolt, either because they are used in engineering to mean other things, or because the common English use of the word does not fit the technical definition so well.  For example, impulse is more commonly used in physics to mean the increase of momentum imparted by a force of limited duration [Belanger 1847], and surge is used by electricians to mean something like an increasing current or voltage.  The terms jerk and jolt are therefore preferred for rate of increase of acceleration.  Jerk appears to be the more common of the two.  It is also recognised in international standards:

In ISO 2041 (1990), Vibration and shock – Vocabulary, page 2:
"1.5 jerk: A vector that specifies the time-derivative of acceleration."
Note that the symbol j for jerk is not in the standard and is probably only one of many symbols used.

As its name suggests, jerk is important when evaluating the destructive effect of motion on a mechanism, or the discomfort caused to passengers in a vehicle.  The movement of delicate instruments needs to be kept within specified limits of both acceleration and jerk to avoid damage.  Engineers who design a train are typically required to keep the jerk less than 2 metres per second cubed for passenger comfort.  The aerospace industry even has a jerkmeter: an instrument for measuring jerk.

In the case of the Hubble space telescope, the engineers are said to have gone as far as specifying limits on the magnitude of the fourth derivative of displacement.  No name for this fourth derivative—the rate of increase of jerk—is univerally accepted.  The term jounce has been used, but has the drawback of using the same initial letter as jerk.  Higher derivatives do not yet have names because they seldom appear in physics.