Equation to describe vehicle position versus time
Equation to describe vehicle position versus time
(OP)
Anyone know if there is a theoretical equation to explain a vehicle's position versus time at full throttle? I'm guessing there are too many variables with tires, wind, etc.
I have plotted several drag race data points on a position versus time graph and curve fit the data. A 5th or 6th degree polynomial seems to be working the best to match the empirical data. The problem is that the typical 1/4 mile drag race only reports time at 5 locations (60', 330', 660', 1000', and 1320'). It also reports velocity at 2 locations (660' and 1320'). When I take the first derivative of the position equation in orderto get a velocity equation, I can't get the reported velocities to match up exactly. I am usually 1 or 2 MPH off. I am trying to use this equation to determine the 0-60mph time.
Now I realize that the instruments may not be calibrated correctly and I also know that with only 5 position data points, my curve fitted equation has some error. So I'm wondering if there is a theoretical equation that is supposed to explain position vs time. If so, I would like to see how well it stacks up to my empirical data.
I am specifically interested in a Dodge Magnum.
I have plotted several drag race data points on a position versus time graph and curve fit the data. A 5th or 6th degree polynomial seems to be working the best to match the empirical data. The problem is that the typical 1/4 mile drag race only reports time at 5 locations (60', 330', 660', 1000', and 1320'). It also reports velocity at 2 locations (660' and 1320'). When I take the first derivative of the position equation in orderto get a velocity equation, I can't get the reported velocities to match up exactly. I am usually 1 or 2 MPH off. I am trying to use this equation to determine the 0-60mph time.
Now I realize that the instruments may not be calibrated correctly and I also know that with only 5 position data points, my curve fitted equation has some error. So I'm wondering if there is a theoretical equation that is supposed to explain position vs time. If so, I would like to see how well it stacks up to my empirical data.
I am specifically interested in a Dodge Magnum.





RE: Equation to describe vehicle position versus time
You might consider doing a spline fit over the segment where you hit 60 mph and interpolate.
TTFN
RE: Equation to describe vehicle position versus time
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RE: Equation to describe vehicle position versus time
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RE: Equation to describe vehicle position versus time
RE: Equation to describe vehicle position versus time
60' 2.004 sec
330' 5.531 sec
660' 8.383 sec 86.13 MPH
1000' 10.834 sec
1320' 12.908 sec 108.55 MPH
RE: Equation to describe vehicle position versus time
6th Degree Polynomial Fit: y=a+bx+cx^2+dx^3...
Coefficient Data:
a = 6.7290643e-013
b = 11.171725
c = 9.2796363
d = 0.16608009
e = -0.071281314
f = 0.0051347964
g = -0.00012727154
The positions matched exactly. Taking the 1st derivative, the velocities are as follows:
8.838 sec 87.94 MPH versus the reported 86.13 MPH
12.908 sec 108.84 MPH versus the reported 108.55 MPH
The 0-60MPH time then comes out to be 4.506 sec.
Greenlight's comments may actually bring the MPH’s a little more in line. Otherwise, I think this is a pretty good fit considering that I only have 5 data points to start with. Any other ideas?
RE: Equation to describe vehicle position versus time
Neglecting shift points, acceleration between shift points should be (roughly) increasing monotonically. A 2nd-order should fit those data points nicely. When you include the shift points, acceleration will drop off sharply for a short time, so you'll have some discontinuities that a 2nd-order won't be able to handle with much grace.
Dan - Owner
http://www.Hi-TecDesigns.com
RE: Equation to describe vehicle position versus time
In any case, the prediction is poor because the discontinuities are not captured in the data. You have a linear term with a relatively large coefficient. This represents an offset in the derivative, which can't possible exist in reality, since you start at t=0 and v=0
I've played around a bit with your data and there doesn't seem to be any way around that minor difficulty, so there isn't any simple polynomial that can accurately capture what you want with so few datapoints.
I would suggest using a spline interpolation and leave it at that. Without using some other instrument with more resolution, you can't get there from here.
TTFN
RE: Equation to describe vehicle position versus time