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I'm dealing with a small toy car consisting of a bottle that has a small hole in the back, covered by a thumb tack. The bottle is filled with pressurized air then the thumb tack is released, releasing a stream of air that acts as a thrust force to propel the car. I am trying to determine its acceleration and force of friction.

I cannot find an equation to determine rolling friction! I do not need an explicit one necessarily, ex. it could be one that could be used to determine it indirectly. Example: If there's an equation that can be used to determine acceleration other than the traditional forces approach (ex. here; I actually thought I had it made with this one, but its derivation for acceleration is on a slope assuming only friction and gravity.) that would enable me to determine net force, then subtract the force of thrust (I've calculated it already) to find the force of friction.

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  • $\begingroup$ What's missing for you in en.wikipedia.org/wiki/Rolling_resistance? $\endgroup$ – CuriousOne Feb 9 '16 at 1:51
  • $\begingroup$ I did see that, but I really need to be as accurate as possible for this project. Most of what I saw seemed generalized for larger cars. Is there anything a bit more specific? $\endgroup$ – Avant Guard Feb 9 '16 at 1:59
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    $\begingroup$ If you want to be accurate you will have to measure it. On the other hand, I think you are probably optimizing for the wrong parameter, to begin with. Rolling resistance with good wheels on a smooth surface will most likely be smaller than the aerodynamic drag and the most important optimization would probably be the nozzle. $\endgroup$ – CuriousOne Feb 9 '16 at 2:07
  • $\begingroup$ I was able to find a pretty good approximation for the thrust force. Could you tell me more about measuring the coefficient of friction or determining aerodynamic drag, though? $\endgroup$ – Avant Guard Feb 9 '16 at 2:28
  • $\begingroup$ EDIT: I wanted to make sure to emphasize this, so I included it separately. Basically, my high school physics teacher is grading me primarily on my calculation of the force of friction and the thrust force. I got the thrust force down, so I just need the friction force now. It doesn't have to be accurate, just well thought out, but I'm lacking in a good method to do such. $\endgroup$ – Avant Guard Feb 9 '16 at 2:30
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Invariably, a good measurement involves comparing something "known" with something "unknown". For example, you put an unknown length next to a ruler.

In your case, you have an "unknown" thrust resulting in some acceleration. You could tie a string to your bottle car, run the string over a pulley, and hang a small weight from it. Measure the acceleration for different weights; compare with the acceleration you get from your thrust. Note that there is a small correction to be made because the system car + weight has slightly more mass than the car by itself, but I think you can work that out. This should tell you what the force of the thrust is - because you know the force of gravity. I am assuming you are able to set this up without friction in the pulley... but note that the inertia due to the mass (and maybe due to the pulley) may also have to be taken into account.

Incidentally, this will also tell you what the force of friction is. You know the force you applied (weight used), the total mass (car plus weight), and the acceleration (I assume you can measure this). In principle, $a=\frac{F}{m}$; but if in practice you find a smaller acceleration than you were expecting, it is because $ a = \frac{F_w - F_f}{m}$ - in other words, it is the net force that is causing the acceleration.

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  • $\begingroup$ You can also measure the distance needed to come to a stop from a known speed and estimate the deceleration as $a = \frac{v^2}{2 x}$. $\endgroup$ – ja72 Mar 12 '18 at 16:37
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You will not find an analytical equation for rolling friction because it depends on so many quantities and unknowns. None of the parts are made with enough accuracy to assume the axle is exactly perpendicular to the wheels, the wheels are exactly round, the center of mass of the wheels exacty on the spin axis and the distribution of material around the wheel to be exactly symmetrical.

As a result, experiments need to be made to estimate an equivalent coefficient of (rolling) friction $$\mu = \frac{a}{g}$$ where $a$ is the deceleration due to rolling friction and $g$ is gravity.

Roll the car without any thrust at a known speed $v$ and measure the distance for it come to a stop. The deceleration $a$ is found by $$a = \frac{v^2}{2 x}$$ where $x$ is the distance to stop.

Do that for various weights of cars (or payloads) and you can see a trend between rolling resistance and wheel loading.

For the next level of sophistication look into the Magic Formula for tires.

$$ R = d \sin \left( c \tan^{-1}\left[(1-e)b k + e \tan^{-1}(b k) \right] \right) $$

To use this formula, one must empirically derive the coefficients with experiments.

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