# Jerk (physics)

In physics, jerk, also known as jolt, surge, or lurch, is the rate of change of acceleration; that is, the derivative of acceleration with respect to time, the second derivative of velocity, or the third derivative of position. Jerk is defined by any of the following equivalent expressions:

$\vec j=\frac {\mathrm{d} \vec a} {\mathrm{d}t}=\frac {\mathrm{d}^2 \vec v} {\mathrm{d}t^2}=\frac {\mathrm{d}^3 \vec d} {\mathrm{d}t^3}$

where

$\vec a$ is acceleration,
$\vec v$ is velocity,
$\vec d$ is position,
$\mathit{t}$ is time.

Jerk is a vector, and there is no generally used term to describe its scalar magnitude (e.g., "speed" as the scalar magnitude for velocity).

The SI units of jerk are metres per second cubed (metres per second per second per second, m/s3, or m·s−3). There is no universal agreement on the symbol for jerk, but j is commonly used. Newton's notation for the derivative of acceleration $(\dot{a})$ can also be used, especially when "surge" or "lurch" is used instead of "jerk" or "jolt".

If acceleration can be felt by a body as the force (hence pressure) exerted by the object bringing about the acceleration on the body, jerk can be felt as the change in this pressure. For example a passenger in an accelerating vehicle with zero jerk will feel a constant force from the seat on his or her body; whereas positive jerk will be felt as increasing force on the body, and negative jerk as decreasing force on the body.

## Applications

Normally concerning forces, speed and acceleration are used for analysis. For example, the "jerk" produced by falling from outer space to the Earth is not particularly useful given the gravitational acceleration changes very slowly. Sometimes the analysis has to extend to jerk for a particular reason.

Jerk is often used in engineering, especially when building roller coasters.[1] Some precision or fragile objects — such as passengers, who need time to sense stress changes and adjust their muscle tension or suffer conditions such as whiplash — can be safely subjected not only to a maximum acceleration, but also to a maximum jerk.[2] Even where occupant safety isn't an issue, excessive jerk may result in an uncomfortable ride on elevators, trams and the like, and engineers expend considerable design effort to minimize it. Jerk may be considered when the excitation of vibrations is a concern. A device that measures jerk is called a "jerkmeter".

Jerk is also important to consider in manufacturing processes. Rapid changes in acceleration of a cutting tool can lead to premature tool wear and result in uneven cuts. This is why modern motion controllers include jerk limitation features.

In mechanical engineering, jerk is considered, in addition to velocity and acceleration, in the development of cam profiles because of tribological implications and the ability of the actuated body to follow the cam profile without chatter.[3]

### Third-order motion profile

In motion control, a common need is to move a system from one steady position to another (point-to-point motion). Following the fastest possible motion within an allowed maximum value for speed, acceleration, and jerk, will result in a third-order motion profile as illustrated below:

The motion profile consists of up to seven segments defined by the following:[4]

1. acceleration build-up, with maximum positive jerk
2. constant maximum acceleration (zero jerk)
3. acceleration ramp-down, approaching the desired maximum velocity, with maximum negative jerk
4. constant maximum speed (zero jerk, zero acceleration)
5. deceleration build-up, approaching the desired deceleration, with maximum negative jerk
6. constant maximum deceleration (zero jerk)
7. deceleration ramp-down, approaching the desired position at zero velocity, with maximum positive jerk

If the initial and final positions are sufficiently close together, the maximum acceleration or maximum velocity may never be reached.

### Jerk systems

A jerk system is a system whose behavior is described by a jerk equation, which is an equation of the form (Sprott 2003):

$\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}= f\left(\frac{\mathrm{d}^2 x}{\mathrm{d} t^2},\frac{\mathrm{d} x}{\mathrm{d} t},x\right)$

For example, certain simple electronic circuits may be designed which are described by a jerk equation. These are known as jerk circuits.

One of the most interesting properties of jerk systems is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations, but which may be combined into a single (although rather complicated) jerk equation.

An example of a jerk equation is:

$\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}+A\frac{\mathrm{d}^2 x}{\mathrm{d} t^2}+\frac{\mathrm{d} x}{\mathrm{d} t}-|x|+1=0.$

Where A is an adjustable parameter. This equation has a chaotic solution for A=3/5 and can be implemented with the following jerk circuit:

In the above circuit, all resistors are of equal value, except $R_A=R/A=5R/3$, and all capacitors are of equal size. The dominant frequency will be $1/2\pi R C$. The output of op amp 0 will correspond to the x variable, the output of 1 will correspond to the first derivative of x and the output of 2 will correspond to the second derivative.

## Non-calculus explanation

Jerk can be difficult to conceptualize when it is defined in terms of calculus. When a force (push or pull) is applied to an object, that object starts to move. As long as the force is applied, the object will continue to speed up. When described in these terms, we are oversimplifying slightly. We think along the lines that there is no force on the object, then suddenly there is a force on the object. We do not think about how long it takes to apply the force.

However, in truth, the application of force does not instantly happen. A change always happens over time. Jerk is the change in acceleration over time. Typically, the time of contact where a force is applied is a split second.

If you push on a wall, it takes a fraction of a second before you apply the full push. Your fingertips will squoosh slightly as you begin to push. How long the squooshing takes determines the jerk. If you push on a wall very slowly, you can actually feel your push increasing. In such a case, the jerk is very low, because the change in force is happening over a relatively long time of several seconds. Jerk happens when a force is applied and removed. But the whole time a force is acting consistently on an object, there is no jerk. (This is because the acceleration is constant when there is a constant force.)

How quickly the force starts its push or pull determines the yank and subsequently the jerk. In most applications, it is not important how quickly the force is applied, and thus we typically think of forces being applied instantaneously. A familiar example of jerk is the rate of application of brakes in an automobile.

An experienced driver gradually applies the brakes, causing a slowly increasing deceleration (small jerk). An inexperienced driver, or a driver responding to an emergency, applies the brakes suddenly, causing a rapid increase in deceleration (large jerk). The sensation of jerk is noticeable, causing the passenger’s head to jerk forward.

### Equations

• $Y = \frac{\Delta F}{\Delta t}$ (yank: force per unit time)
• $j = \frac{\Delta a}{\Delta t}$ (jerk: acceleration per unit time)
• $j = \frac{y}{m}$, following from Newton's second law divided by $\Delta t$ and the above two relations: $\frac{\Delta F}{\Delta t} = m \frac{\Delta a}{\Delta t} \implies y = m j$
• $j = \frac{\Delta F}{m \Delta t}$

The higher the force or acceleration, the higher the jerk. The shorter the time of change in acceleration, such as a rollercoaster 'whipping' around a corner, the higher the jerk. For uniform jerk, the following equation can be applied:

• $a = a. + jt$
• $v = u + a.t + \frac{1}{2}jt^2$
• $s = ut + \frac{1}{2}a.t^2 + \frac{1}{6}jt^3$

where a : final acceleration a. : initial acceleration j : jerk (change in acceleration) v : final velocity u : initial velocity s : distance/displacement t : time taken