Why does rubber shoot faster on a hot day and slower on a cold day?
Natural rubber uses heat to contract. The more heat there is, the more it will contract. This can be shown as follows:
If you heat a band under load, it will contract.
Also, its resilience, which is the major input in band efficiency (the difference in the elastic potential energy you put in and the kinetic energy you get out) increases under heat.
I won't bore you with the derivation of the entropy equation, but in simple words, the effect of temperature is proportional. That is to say the force and resilience is proportional to change in temperature and specifically absolute temperature (degrees above absolute zero) at least for temperatures above brittle temperature and for elongations below the inelastic region.
Further notes: Tex-Shooter first pointed this effect out to me when I started looking into rubber. Thank you Bill. Bill also asked my to look into why thin bands should be better than thick bands on cold days. I have no explanation for his observation except to say that if a thin taper is drawn into its inelastic region, then the contractile force of elastic drawn into the inelastic region isn't purely down to the temperature effects of the elastomer, but the contractile force per unit extension will rise as the band becomes inelastic. In this transition into inelasticity, the number of molecules available to continue to be stretched decreases and the band's resilience is less dependent on temperature and the limits of inelasticity take over as the more important factor. Although an over-stretched band will be less efficient in terms of energy that comes out vs energy that goes in, much more energy is loaded into the over stretched band compared to the fixed inertia of the band, pouch and projectile, so it shoots faster.
I am quite surprised this week, at the effect the colder temperatures are having on my shooting. I find I have had to adjust aiming point to compensate for the lower temperatures.
Sorry, I used to use convert all my figures to Imperial units before publishing, but it gets totally out of hand when dealing with thermodynamics.
The key message from the chart is it's a linear (straight line relationship) at all temperatures between just above the freezing and boiling points of water.
270°K = 26°F
290°K = 62°F
310°K = 98°F
330°K = 134°F
350°K = 170°F
370°K = 206°F
390°K = 242°F
However, as you can see from the chart, it's not quite as simple as dividing one temperature number by another to calculate the difference in force. Tensile force doesn't go to zero at 'zero' on the Fahrenheit scale, the freezing point of water, or the brittle point of elastic, but being a thermodynamic process, it intercepts the Y-axis at absolute zero. Note however, that at around the brittle point of rubber, rubber becomes totally inelastic and simply cracks and snaps. Above a certain point, it melts down or chars.
...
I previously kept formulas out of this to avoid confusion, but FYR, the key formula governing static tensile force and temperature is:
where:
f = Force in Newtons. One Newton is about 3.6 International avoirdupois ounces of force
A0 = unstretched cross sectional area in square metres. One m2 is about 10.8 sqft.
N = Number of chain segments per cubic metre. One cubic metre is about 35.3 cubic feet
T = temperature in degrees Kelvin. Use Hrawk's formulae or my table above to convert into Imperial units.
k is the Boltzmann constant in J/K. One Joule is 2.39 × 10^−4 kcal
Lambda is the ratio of the length of the stretched to unstretched band
and note:
N is a constant that depends on the composition of the rubber and its freshness. One experiment I have read suggests that it is around 2.2 × 10^25 for natural rubber, but personally I find that this figure's a bit low.
The Boltzmann constant is 1.38 x 10^-23 J/K
Ignore all the constants and conversion factors and just consider how the key elements of the equation work together:
The force required for an extension is proportional to (absolute) temperature ... as has been shown above.
The force required for an extension is proportional to the cross sectional area of the band.
The force required for an extension is not a straight line but a curve.
A picture is worth a thousand words so these two charts are worth about 2,000 words in total:
This chart above predicts the force-elongation curve of rubber very well. It shows that the 'knee' at low elongations aren't really due to a yielding of the material, but it just down to the nature of rubber. The relationship then becomes more or less proportional. Algebraically, it says that at higher elongations, the relationship is proportional to the elongation ratio Lambda as the term 1/(Lambda squared) becomes vanishingly small with greater elongations). This relationship breaks down at very high elongations as the number of available molecular connections that have not already reached full extension runs out. It's not abrupt because the elastomer molecules have a random number of connections. You can use Gaussian statistics to model this process but unless you have a particular fetish for greeks and integrals, you won't be interested and if you do, you'll already have gottent the point. Rubber gradually develops inelasticity somewhere at the right hend side of the curve and the maount of force needed to get more draw goes up. I'll go into detail on this topic in another post.
The chart above tells you how to cut your bands to get a certain increase of force out of a certain greater width of band. Double the force? Double the width. Reduce the force by a fifth? Reduce the width by a fifth. No width of band = no force required.
Thanks for another great blog Z.I am quite surprised this week, at the effect the colder temperatures are having on my shooting. I find I have had to adjust aiming point to compensate for the lower temperatures.
Yes, there seems to be a greater effect on velocity than the chart would suggest.
I think I may have to look into the temperature thing again.
Here is teaching materials that shows the relationship I have shown in my original chart (It also hints that you should try urethanes if you want consistency, but the contractile force per mass is not given, so urethanes may prove to be slower even if they are more consistent).
Now, I understand 'resilience' to mean the ratio of contractile force under unloading over the contractile force under loading, so it is possible that this effect is on top of the thermodynamic process given in the formula in my comment above. That would mean the effect is doubled! Remember also that the resilience is a dynamic effect and contractile force is measured at equilibrium. I would appreciate anyone that has good data or can help understand the text of the link above.
Another reason why the drop may be more affected than is indicated in the chart might be due to the way energy (force x distance on unloading) is related to velocity. If the projectile is small compared to the mass of the bandset and pouch, then the change in velocity will be more than the change in force. A lower velocity will make the flight time longer and remember that gravity makes the ball accelerate downwards and that means a reduction in velocity will result in more than a proportional increase in projectile drop.
So the chart of resilience vs force above doesn't directly relate to how you should adjust your aim. You'd need a very sophisticated model for that; the kind of thing you would submit as a thesis. The most efficient way to adjust your aim remains the time-honoured method of trial and error.
I performed a small and very unscientific test this morning on the effects of temperature.
First thing this morning with a temp of approx 11 degrees C, shooting Fastbands, I got my point of aim right for hitting a can at around 8m.
About 2 hours later, with my bands having sat in the sun (wrapped in a black cloth bag), I used the same point of aim and took a few more shots. The ambient temp is up to 18c and I estimate that the band temp was around 30c.
For the first few shots, each shot was consistently 12-15cm high with a notable speed increase and flatter trajectory. After around 20 shots, It eventually settled to around 4cm high as the bands came back to the ambient temp of 18c.
When I'm back home next week, I'll perform the same tests with accurate temp measurements and chrony results.
Anyhow, keep up the good work Z, I hope it isn't giving you too much of a headache
I think you may be looking for a polyurethane rubber. Urethanes are less resilient than natural rubber (shoot slower) but the resilience is less affected by temperature, so it is possible that they may shoot faster at winter temperatures.
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