# How do I calculate the "overall" jerk of my motion stimuli (2D coordinates per frame), kind of a "medium jerk"?

## I'm a psychologist and I'm currently studying physics but, because this is not my field of study, I'm having great difficulties. I'm following the physics movies here and i'm learning a lot (thanks a lot!). I am currently studying motion and I have to calculate the jerk of my 2D motion data and I don't even know where to start. My data is basically a bunch of coordinates on the x and y axis in time. I'll be using jerk as a measure of the "quantity of acceleration" of my stimuli. The aim is to compare the different jerk values of each of my stimulus. I have the velocity calculated per frame of my stimuli (i.e. I have a velocity value per time - because time is constant). I used this formula for calculating the velocity: =SQRT((X2-X1)^2+(Y2-Y1)^2)/1/60), where X1 is the initial position of the stimulus on the x axis and X2 is the following position and Y1 is the initial position of the Y axis and Y2 is the second position of the Y axis. This is data captured at 60Hz, thus, time variation is constant and is 1/60. Thus, I have the instantaneous velocity per frame calculated. However, I want to calculate the "overall" jerk of my stimuli, kind of a "medium jerk"? How do I do that? Any help would be truly important! Thanks a lot!

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Jan 11, 2017

This question has not been answered here for almost one year. I will attempt to place my ideas which are as below.

It appears to be a good idea to use the concept of jerk as understood in motion to assess the jerk associated with psychological stimuli.

Before you do so you need to understand that
A
1. Displacement, 2. Velocity 3. Acceleration, 4. Jerk are all vector quantities. This means that each of above has a magnitude and direction attached to it.

Whether psychological stimuli can be considered to be vector quantities or not needs to be understood and stated ab-initio for making a one-to-one comparison.

B. As per definition
Displacement $\vec{s}$ is the position vector of a point with respect to a coordinate system.

Velocity $\vec{v}$ is rate of change of displacement with time. It is first differential of the position vector with time.

$\vec{v} \equiv \frac{\mathrm{dv} e c s}{\mathrm{dt}}$

Acceleration $\vec{a}$ is rate of change of velocity with time. It is first differential of the velocity vector with time or second differential of displacement vector with time.

$\vec{a} \equiv \frac{\mathrm{dv} e c v}{\mathrm{dt}} = \frac{{d}^{2} \vec{s}}{\mathrm{dt}} ^ 2$

Jerk $\vec{j}$ is rate of change of acceleration with time. It is first differential of the acceleration vector with time or second differential of velocity vector with time or third differential of displacement vector with time.

$\vec{j} \equiv \frac{\mathrm{dv} e c a}{\mathrm{dt}} = \frac{{d}^{2} \vec{v}}{\mathrm{dt}} ^ 2 = \frac{{d}^{3} \vec{s}}{\mathrm{dt}} ^ 3$

In any equation the vector could be constant or changing with time. For constant vectors which do not change with time, first differential is always zero.
The change could be constant or time dependent. For a constant change, as already stated above first differential is zero whereas for time dependent vector changes first differential will have either constant value or time dependent value.

What you have calculated with the expression
velocity $= \frac{\sqrt{{\left({X}_{2} - {X}_{1}\right)}^{2} + {\left({Y}_{2} - {Y}_{1}\right)}^{2}}}{60}$ is the scalar or modulus part of the velocity or average speed.

C.
It will a good idea to start plotting a number of observations in time interval of interest on a graph. The graph will tell you the characteristics of your variable.

For example, a straight line graph between position vector and time indicates, constant or zero velocity (depending on the slope), zero acceleration and zero jerk as shown in the figure below, red and green graphs.

To calculate the value of jerk of stimuli an acceleration-time graph will be required.

Alternatively, for a ${t}^{2}$ dependence on distance (depicted by blue line in the figure above), jerk is zero. As such for a finite values of jerk we need ${t}^{3}$ or higher power dependence of displacement and time.

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