From Position to Jerk
Position describes where an object is. Its first derivative with respect to time describes velocity, and its second derivative describes acceleration. The next derivative, the third derivative of position, measures how quickly acceleration changes. This quantity is called jerk.
Jerk is useful when a changing acceleration matters. For example, a vehicle can have the same acceleration at two moments but feel different if its acceleration is increasing at one moment and decreasing at the other. Jerk describes that change. It does not, by itself, tell us whether the object is moving forward or speeding up.
Here, \(s\), \(v\), \(a\), and \(j\) are functions of time. The notation \(s'''(t)\) means the third derivative of position with respect to time. Jerk is an instantaneous rate of change: at a specified time, it tells how quickly acceleration is changing at that instant.
Why the Third Derivative Gives Jerk
The definition follows from the meaning of a derivative. The derivative of acceleration is the limit of the average change in acceleration over a short time interval. Since acceleration is already the second derivative of position, differentiating it once more gives the third derivative of position.
This argument requires that \(s''\) be differentiable at the time in question—that is, that the third derivative \(s'''(t)\) exist. If acceleration has a sharp jump or is not differentiable at a time, jerk may not be defined there, even if position and acceleration have meaningful values.
Units follow from taking derivatives with respect to time. If position is measured in meters and time in seconds, each derivative divides the units by one additional second.
The unit \(\mathrm{m/s^3}\) means meters per second cubed. It is not a position or a velocity; it is a rate of change of acceleration. In other unit systems, the same pattern applies: the units of position are divided by time once for each derivative.
Worked Example: Jerk from a Polynomial Position Function
Worked Example: Jerk from a Polynomial Position Function
An object's position, in meters, after \(t\) seconds is \(s(t)=2t^3-9t^2+12t+1\). Find its velocity, acceleration, and jerk, and evaluate them at \(t=2\) seconds.
Differentiate position once to find velocity:
Differentiate velocity to find acceleration:
Differentiate acceleration to find jerk:
Because position is in meters and \(t\) is in seconds, the units are \(\mathrm{m/s}\) for velocity, \(\mathrm{m/s^2}\) for acceleration, and \(\mathrm{m/s^3}\) for jerk. At \(t=2\) seconds:
Check the jerk by differentiating the acceleration formula \(a(t)=12t-18\): its derivative is \(12\), as found. At \(t=2\), acceleration is \(6\ \mathrm{m/s^2}\) and jerk is positive, so acceleration is increasing at that instant. The zero velocity at that instant does not make the jerk zero.
Worked Example: Jerk from Sinusoidal Position
Worked Example: Jerk from Sinusoidal Position
Suppose an object's position is \(s(t)=0.5\cos(2t)\) meters, where \(t\) is measured in seconds and the angle \(2t\) is in radians. Find the velocity, acceleration, and jerk, then evaluate them at \(t=\pi/4\) seconds.
Apply the chain rule at each differentiation. For velocity, the derivative of \(\cos(2t)\) is \(-\sin(2t)\cdot 2\):
Differentiate velocity, again multiplying by the derivative of the inside expression \(2t\):
Differentiate acceleration to obtain jerk:
At \(t=\pi/4\), the angle is \(2t=2(\pi/4)=\pi/2\). Since \(\sin(\pi/2)=1\) and \(\cos(\pi/2)=0\):
As a check, differentiating the displayed acceleration formula gives \(a'(t)=4\sin(2t)\), which agrees with the jerk formula. At this instant, acceleration is zero but jerk is positive: the acceleration is changing through zero, rather than remaining zero.
Higher Derivatives of Position
After the third derivative, the pattern continues as long as the required derivatives exist. The fourth derivative of position is the derivative of jerk. It is sometimes given the informal name snap, though terminology for this derivative varies. For clarity, it is enough to call it the fourth derivative of position.
For position measured in meters and time measured in seconds, the fourth derivative has units of \(\mathrm{m/s^4}\). The same unit rule continues: every additional time derivative introduces one more factor of \(1/\mathrm{s}\). A higher derivative is not automatically more important in every motion problem; whether it is useful depends on what aspect of the motion is being modeled.
Worked Example: Jerk and the Fourth Derivative
Consider the position function \(s(t)=(1\ \mathrm{m/s^4})t^4-(6\ \mathrm{m/s^3})t^3+(12\ \mathrm{m/s^2})t^2\), with \(t\) in seconds. Find jerk and the fourth derivative, then evaluate both at \(t=1\) second.
Differentiate term by term. The coefficients include units so that each term in position has units of meters.
At \(t=1\) second:
Check by differentiating the jerk expression: the derivative of \((24\ \mathrm{m/s^4})t-36\ \mathrm{m/s^3}\) is \(24\ \mathrm{m/s^4}\), matching the fourth derivative. At one second, jerk is negative, so acceleration is decreasing at that instant.
Interpreting Jerk Carefully
The sign of jerk describes the direction in which acceleration is changing. Positive jerk means acceleration is increasing; negative jerk means acceleration is decreasing; and zero jerk means acceleration has zero instantaneous rate of change, if jerk exists. These statements concern acceleration, not directly the object's direction of motion.
For instance, positive jerk does not necessarily mean the object is moving in the positive direction. The object could have negative velocity while its acceleration is increasing. Likewise, zero jerk at one instant does not prove that acceleration is constant over an entire interval; it only says that the instantaneous derivative of acceleration is zero at that instant.
Identify the position function and the time interval or instant of interest.
Find velocity, then acceleration, then jerk. For a fourth derivative, differentiate jerk once more.
Divide the position units by the time units once for each derivative.
Jerk describes the instantaneous change in acceleration, not the velocity or acceleration value itself.
A Common Misreading
A frequent error is to treat jerk as another name for acceleration, or to interpret positive jerk as positive acceleration. These are different quantities. For example, in the polynomial example above, \(a(2)=6\ \mathrm{m/s^2}\) and \(j(2)=12\ \mathrm{m/s^3}\). The first value tells the acceleration at that instant; the second tells how that acceleration is changing.
A related calculation mistake is to stop after finding \(s''(t)\) when a question asks for jerk. That gives acceleration, not jerk. The correction is to differentiate once more: \(s(t)\) to \(s'(t)\), then to \(s''(t)\), then to \(s'''(t)\). In a trigonometric model, apply the chain rule at every differentiation, including the derivatives of the inner expressions.
Why Higher Motion Derivatives Matter
Acceleration describes changes in velocity, while jerk describes changes in acceleration. In applications where abrupt changes in acceleration are important, jerk provides information that position, velocity, and acceleration alone do not convey. It also gives a way to compare motion models: two objects can have equal acceleration at a particular time but different jerk at that time.
For AP Calculus problems, the main skills are to identify the requested derivative, compute it accurately, evaluate it at the correct time, and state its units and meaning. The derivative order and the units should agree: jerk is a third derivative of position and has position-units divided by time cubed.
Check Your Understanding
Answer each question using the definitions and methods in this tutorial.
- If position is measured in feet and time in seconds, what are the units of jerk?
- For \(s(t)=t^3-3t^2\), find the jerk function and state whether it is positive, negative, or zero.
- If \(a(t)=5-2t\), what is the jerk, and what does its sign say about acceleration?
- For \(s(t)=\cos(3t)\), differentiate three times to find the jerk function.
- Can an object have zero acceleration and nonzero jerk at the same instant? Explain briefly.
