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Jerk
Time-derivatives of position, including jerk
Common symbols	j, j, ȷ→
In SI base units	m/s3
Dimension	L T−3

In physics, jerk (also known as jolt) is the rate of change of an object's acceleration over time. It is a vector quantity (having both magnitude and direction). Jerk is most commonly denoted by the symbol j and expressed in m/s³ (SI units) or standard gravities per second (g₀/s).

Expressions

As a vector, jerk j can be expressed as the first time derivative of acceleration, second time derivative of velocity, and third time derivative of position:

Where:

• a is acceleration
• v is velocity
• r is position
• t is time

Third-order differential equations of the form

are sometimes called jerk equations. When converted to an equivalent system of three ordinary first-order non-linear differential equations, jerk equations are the minimal setting for solutions showing chaotic behaviour. This condition generates mathematical interest in jerk systems. Systems involving fourth-order derivatives or higher are accordingly called hyperjerk systems.^[1]

Physiological effects and human perception

Human body position is controlled by balancing the forces of antagonistic muscles. In balancing a given force, such as holding up a weight, the postcentral gyrus establishes a control loop to achieve the desired equilibrium. If the force changes too quickly, the muscles cannot relax or tense fast enough and overshoot in either direction, causing a temporary loss of control. The reaction time for responding to changes in force depends on physiological limitations and the attention level of the brain: an expected change will be stabilized faster than a sudden decrease or increase of load.

To avoid vehicle passengers losing control over body motion and getting injured, it is necessary to limit the exposure to both the maximum force (acceleration) and maximum jerk, since time is needed to adjust muscle tension and adapt to even limited stress changes. Sudden changes in acceleration can cause injuries such as whiplash.^[2] Excessive jerk may also result in an uncomfortable ride, even at levels that do not cause injury. Engineers expend considerable design effort minimizing "jerky motion" on elevators, trams, and other conveyances.

For example, consider the effects of acceleration and jerk when riding in a car:

• Skilled and experienced drivers can accelerate smoothly, but beginners often provide a jerky ride. When changing gears in a car with a foot-operated clutch, the accelerating force is limited by engine power, but an inexperienced driver can cause severe jerk because of intermittent force closure over the clutch.
• The feeling of being pressed into the seats in a high-powered sports car is due to the acceleration. As the car launches from rest, there is a large positive jerk as its acceleration rapidly increases. After the launch, there is a small, sustained negative jerk as the force of air resistance increases with the car's velocity, gradually decreasing acceleration and reducing the force pressing the passenger into the seat. When the car reaches its top speed, the acceleration has reached 0 and remains constant, after which there is no jerk until the driver decelerates or changes direction.
• When braking suddenly or during collisions, passengers whip forward with an initial acceleration that is larger than during the rest of the braking process because muscle tension regains control of the body quickly after the onset of braking or impact. These effects are not modeled in vehicle testing because cadavers and crash test dummies do not have active muscle control.
• To minimize the jerk, curves along roads are designed to be clothoids as are railroad curves and roller coaster loops.

Force, acceleration, and jerk

For a constant mass m, acceleration a is directly proportional to force F according to Newton's second law of motion:

In classical mechanics of rigid bodies, there are no forces associated with the derivatives of acceleration; however, physical systems experience oscillations and deformations as a result of jerk. In designing the Hubble Space Telescope, NASA set limits on both jerk and jounce.^[3]

The Abraham–Lorentz force is the recoil force on an accelerating charged particle emitting radiation. This force is proportional to the particle's jerk and to the square of its charge. The Wheeler–Feynman absorber theory is a more advanced theory, applicable in a relativistic and quantum environment, and accounting for self-energy.

In an idealized setting

Discontinuities in acceleration do not occur in real-world environments because of deformation, quantum mechanics effects, and other causes. However, a jump-discontinuity in acceleration and, accordingly, unbounded jerk are feasible in an idealized setting, such as an idealized point mass moving along a piecewise smooth, whole continuous path. The jump-discontinuity occurs at points where the path is not smooth. Extrapolating from these idealized settings, one can qualitatively describe, explain and predict the effects of jerk in real situations.

Jump-discontinuity in acceleration can be modeled using a Dirac delta function in jerk, scaled to the height of the jump. Integrating jerk over time across the Dirac delta yields the jump-discontinuity.

For example, consider a path along an arc of radius r, which tangentially connects to a straight line. The whole path is continuous, and its pieces are smooth. Now assume a point particle moves with constant speed along this path, so its tangential acceleration is zero. The centripetal acceleration given by v²/r is normal to the arc and inward. When the particle passes the connection of pieces, it experiences a jump-discontinuity in acceleration given by v²/r, and it undergoes a jerk that can be modeled by a Dirac delta, scaled to the jump-discontinuity.

For a more tangible example of discontinuous acceleration, consider an ideal spring–mass system with the mass oscillating on an idealized surface with friction. The force on the mass is equal to the vector sum of the spring force and the kinetic frictional force. When the velocity changes sign (at the maximum and minimum displacements), the magnitude of the force on the mass changes by twice the magnitude of the frictional force, because the spring force is continuous and the frictional force reverses direction with velocity. The jump in acceleration equals the force on the mass divided by the mass. That is, each time the mass passes through a minimum or maximum displacement, the mass experiences a discontinuous acceleration, and the jerk contains a Dirac delta until the mass stops. The static friction force adapts to the residual spring force, establishing equilibrium with zero net force and zero velocity.

Consider the example of a braking and decelerating car. The brake pads generate kinetic frictional forces and constant braking torques on the disks (or drums) of the wheels. Rotational velocity decreases linearly to zero with constant angular deceleration. The frictional force, torque, and car deceleration suddenly reach zero, which indicates a Dirac delta in physical jerk. The Dirac delta is smoothed down by the real environment, the cumulative effects of which are analogous to damping of the physiologically perceived jerk. This example neglects the effects of tire sliding, suspension dipping, real deflection of all ideally rigid mechanisms, etc.

Another example of significant jerk, analogous to the first example, is the cutting of a rope with a particle on its end. Assume the particle is oscillating in a circular path with non-zero centripetal acceleration. When the rope is cut, the particle's path changes abruptly to a straight path, and the force in the inward direction changes suddenly to zero. Imagine a monomolecular fiber cut by a laser; the particle would experience very high rates of jerk because of the extremely short cutting time.

In rotation

Consider a rigid body rotating about a fixed axis in an inertial reference frame. If its angular position as a function of time is θ(t), the angular velocity, acceleration, and jerk can be expressed as follows:

• Angular velocity, ${\displaystyle \omega (t)={\dot {\theta }}(t)={\frac {\mathrm {d} \theta (t)}{\mathrm {d} t}}}$, is the time derivative of θ(t).
• Angular acceleration, ${\displaystyle \alpha (t)={\dot {\omega }}(t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}}$, is the time derivative of ω(t).
• Angular jerk, ${\displaystyle \zeta (t)={\dot {\alpha }}(t)={\ddot {\omega }}(t)={\overset {...}{\theta }}(t)}$, is the time derivative of α(t).

Angular acceleration equals the torque acting on the body, divided by the body's moment of inertia with respect to the momentary axis of rotation. A change in torque results in angular jerk.

The general case of a rotating rigid body can be modeled using kinematic screw theory, which includes one axial vector, angular velocity Ω(t), and one polar vector, linear velocity v(t). From this, the angular acceleration is defined as

and the angular jerk is given by

taking the angular acceleration from Angular acceleration#Particle in three dimensions as

, we obtain