I have speed data from the GPS transmitter of a Truck which reports the speed of the vehicle at a fixed time interval. I can calculate the acceleration/deceleration of the truck by doing a $\frac{v_2-v_1}{t_2-t_1}$ calculation. However, can I calculate the jerk in the same manner? That is $\frac{a_2-a_1}{t_2-t_1}$? Will that be a correct thing to do?
[Physics] How to calculate the jerk from acceleration data
differentiationjerkkinematics
Related Solutions
I'm fairly sure it experiences an acceleration in order to change direction.
Yes, any change in velocity means an acceleration has occurred. That includes a change in direction of the velocity.
Since the object experiences a change in direction at some point but still travels at the same speed, would the second velocity technically be negative?
If the first velocity is in the positive direction, yes the second might be negative. Is this a 1-dimensional problem?
I want to use the formula $A_x=\Delta v/ \Delta t.$
That will give you the average acceleration over the total time. It tells you nothing about the instantaneous acceleration at any specific time. Also, for $A_x$ you would use $\Delta v_x$.
Any curvilinear motion (which can't happen in 1-D motion) requires a sideways acceleration component of $$ a_{\perp} = \frac{v^2}{r} $$ where $v$ is the instantaneous speed and $r$ is the instantaneous radius of curvature.
There are various ways of calculating acceleration and jerk from your data set.
The value of $\mathrm{d}t$ you want is the time step generally, in your case $100 \, \mathrm{ms}.$
The method that you suggest $$a=\frac{v_2-v_1}{\mathrm{d}t}$$ will give you an answer and is reasonable, but it will not give you the best value I expect. It might be better to use something like Savitsky-Golay fitting of the data set, which you can do to get derivatives.
The formula you suggest can be used for jerk, just put in two accelerations instead of two velocities. The only problem with this is that at each step you are getting further and further from the data, so any experimental errors in position will get magnified in each step of calculation for velocity, acceleration and the worst would be jerk.
I would be tempted to use two different methods and finding velocity (+ acceleration and jerk if possible) and then compare the velocities from different methods and it may give an indication of how reliable your analysis is.
The Wikipedia page on Savitsky-Golay fitting has parameters for finding 1st 2nd 3rd... derivatives at the bottom.
Note that in reply to your question I would draw up a table in each dimension like this:
$$ \begin{array}{c|c|c|c|c} \text{time,}~t & \text{position,}~x & v=\frac{\mathrm{d}x}{\mathrm{d}t} & a=\frac{\mathrm{d}^2x}{{\mathrm{d}t}^2} & \text{jerk}=\frac{\mathrm{d}^3x}{{\mathrm{d}t}^3} \\ \hline 0 & 0 & 10 & 5 & 4 \\ 1 & 10 & 15 & 9 & 3 \\ 2 & 25 & 24 & 12 & 2 \\ 3 & 49 & 36 & 14 & 1 \\ 4 & 85 & 50 & 15 & \vdots \\ 5 & 135 & 65 & \vdots \\ 6 & 200 & \vdots \\ \vdots & \vdots \end{array}_{\Large{.}} $$
Hope this is helpful – of course the velocities should have times in between the times for $x$ and the accelerations times between the times of the velocities etc. etc. – but it is not so easy to figure out how to indicate that in a table in an answer here.
Best Answer
That is correct. The jerk is the 3'rd derivative of position with respect to time, which is the change in acceleration per unit time. Keep in mind that position, velocity, acceleration, and jerk are vectors. Your formula would compute the magnitude of the jerk. To compute its vector, you would use your formula and treat the acceleration as vectors.