Postion -> Velocity -> Acceleration -> Jerk ->?

In summary, for a function x(t) representing position as a function of time, the first derivative x'(t) represents velocity, the second derivative x''(t) represents acceleration, and the third derivative x'''(t) represents jerk. The fourth derivative x''''(t) is sometimes referred to as a 'spasm' or 'jounce' and can be calculated using a position/time graph. It is often used to find maximum and minimum accelerations. Higher order derivatives, such as fifth and sixth, may also be useful in certain scenarios, such as for harmonic functions like a pendulum.
  • #1
tectactoe
39
0
We all know that, assuming [tex]x(t) =[/tex] position as a function of time, then:

[tex]x'(t) = v(t) = velocity[/tex]
[tex]x''(t) = v'(t) = a(t) = acceleration[/tex]
[tex]x'''(t) = v''(t) = a'(t) = j(t) = jerk[/tex] (assuming j is the symbol for jerk).

But what does [tex]x''''(t) = j'(t)[/tex] come out to be? Is there a fourth derivative of position? And if so, is it ever practically used?

What about fifth, sixth, seventh, etc derivatives?

This is just something I've been extremely curious about since I learned of the third derivative, jerk (or jolt).

Thanks!
 
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  • #2
The term ¨x¨[d4x/dt4 is the time derivative of the jerk,
which might be called a ‘‘spasm.’’ It has also been called a
‘‘jounce,’’ a ‘‘sprite,’’ a ‘‘surge,’’ or a ‘‘snap,’’ with its successive
derivatives, ‘‘crackle’’ and ‘‘pop.’’

http://sprott.physics.wisc.edu/pubs/paper229.pdf

'snap', 'crackle' and 'pop'...:smile:
 
  • #3
Haha that's funny.

Does anyone happen to have a position/time graph in which you'd be able to calculate something like the fifth or sixth derivative of x? Would these ever even be needed? Hah.
 
  • #4
Anything that is a harmonic function, like say a pendulum, will have a non zero nth order x^n derivative thing.

say, x = Sin(t)
 
  • #5
Derivatives of order 4 & 5 can be used to find and classify the maximum/minimum accelerations, for example.
 

Related to Postion -> Velocity -> Acceleration -> Jerk ->?

1. What is the relationship between position, velocity, acceleration, jerk, and time?

The relationship between these quantities is described by the fundamental laws of motion, specifically Newton's second law. Position is the location of an object in space, velocity is the rate of change of position over time, acceleration is the rate of change of velocity over time, and jerk is the rate of change of acceleration over time. Time is the independent variable that ties all of these quantities together.

2. How are position, velocity, acceleration, and jerk measured and represented in physics?

In physics, position is typically measured and represented using a coordinate system, such as Cartesian coordinates. Velocity is measured and represented as the change in position over time, typically in units of meters per second. Acceleration is measured and represented as the change in velocity over time, typically in units of meters per second squared. Jerk is measured and represented as the change in acceleration over time, typically in units of meters per second cubed.

3. What is the physical significance of jerk in relation to position, velocity, and acceleration?

Jerk is the rate of change of acceleration over time, and it represents the amount of force or impact experienced by an object as it moves. High jerk values can result in discomfort or injury for living organisms, while low jerk values can indicate smooth, efficient motion.

4. Are there real-world examples of objects or systems that demonstrate the concept of jerk?

Yes, there are many real-world examples of objects or systems that demonstrate jerk. For example, a car accelerating and decelerating rapidly experiences high jerk values, while a smooth rollercoaster ride would have lower jerk values. Similarly, a person swinging on a swing set experiences jerk at the bottom of the swing as they change direction, and a person riding a bike over a bumpy road experiences jerk as they hit the bumps.

5. How is the concept of jerk applied in fields other than physics?

The concept of jerk can also be applied in fields such as engineering, biomechanics, and robotics. In engineering, jerk is an important consideration in designing smooth and efficient motion for machines. In biomechanics, jerk is used to understand the forces and stresses on the human body during movement. In robotics, jerk is used to improve the accuracy and precision of robotic movements.

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