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    some ballistics and bow power calculations

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    phuphuphnik
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    some ballistics and bow power calculations Empty some ballistics and bow power calculations

    Post by phuphuphnik on Fri Feb 21, 2014 5:49 am

    This is a work in progress for the 7th grade stonebow project. I think that many of you will be able to use this. I'll check out the experimental results and see how they line up with the calculations. Right now the 217 FPS I came up with seems a bit optimistic, so a measurement may be off.
    Cheers,
    chriso



    Measure the angle between the spanned and drawn bow, with the point where the string connects to the prod making the base, and the drawn bowstring making one side. This angle is α. Next measure the angle of where the drawn string is held by the nut. This is β. You will also need the draw weight of the bow. This is W.
     The equation is:     W=2F(cosα)(cosβ)
    For this bow W is 70, α is 42°, and β is 58°. Working it out** we get:
    70=2F(cos42°)(cos58°)
    70=2F(.74)(.53)
    70=2F×.3938
    70/.3938=2F
    177.75=2F
    177.75/2=F
    88.8=F
    The Force is 88.8 LbF (pounds of force) We’ll need to change that to the SI unit of
    Newtons (N) by multiplying it by 4.448, so 88.8×4.448=394.98 Newtons.
    Once you have the force in Newtons, you can figure out the energy in Joules.
    E=FX÷2
    E in energy in Joules, F is the Force in Newtons, and X is the draw length in Meters, in this case 6” which is 15.5 cm, or .155 meters.
    E=(395×.155)/2=30.61 Joules
    E=30.61 Joules.
    Now we can calculate velocity. E=1/2 MV^2 E is 30.61, M is the mass of the projectile, in this case 15 grams, or .015 Kg and V is in Meters per second.
    30.61=1/2 (.015)V^2
    30.61=.007V^2
    30.61/.007=V^2
    4372.8=V^2
    √4372.8=V
    66.13=V
    The projectile will have a calculated velocity of 66.13 M/s or ±217 FPS

    We need to remember that this does not account for drag/wind resistance, or friction from the bow channel or other factors like this. It will however, give you a pretty good approximation of what to expect. This also works best with a plain old curved bow, recurves get more tricky, and this won’t work with compound bows.

    ** Make sure your calculator is set to DEG not RAD (radians) or the whole thing will be screwy. On the Google calculator the toggle is on the upper left just under the number line. On most calculators there is a button, and the display will say DEG or RAD.

    Now, to test this experimentally.
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    Post by Hotspur on Fri Feb 21, 2014 7:24 am

    I'm a little math challenged but your speed sounds reasonable.  Do you have a chronograph?
     
    Your prod will also have an optimal projectile weight for efficiently utilizing its potential energy and delivering the greatest force to the target, calculated as approximately 1/12th the mass of the prod.

    I fiddled around with bolt mass in this thread http://thearbalistguild.forumotion.com/t1029p15-rules-of-thumb-on-the-optimal-bolt-weight

    I measured the average bolt speed over 50' as I don't have a chronograph. 

     
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    Post by Hotspur on Fri Feb 21, 2014 7:29 am

    Also, 15 grams is very light (231 gns).  Is there a risk of dry fire damage here?  Others may know more about that here...
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    Post by hullutiedemies on Fri Feb 21, 2014 9:44 am

    What are you doing with those angles?
    I am not sure if I understand your description, but it seems that you are computing string tension. Why ?
    You state that you allready know the draw weight. This is the accelerating force.
    70#/6" bow should store ca. 25J assuming linear F/D-curve.

    To get any sort of reliable estimate for speed you need to know total inertia you are accelerating. That is the mass of projectile plus the inertia momentum of limbs and string.
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    Post by phuphuphnik on Fri Feb 21, 2014 11:39 pm

    Nerd Flintstone wrote:
    To get any sort of reliable estimate for speed you need to know total inertia you are accelerating. That is the mass of projectile plus the inertia momentum of limbs and string.

    This is why I'm using the angles. Using Hooke's law I can begin showing the students the relationship between the distance pulled back and the power imparted.
    This site: http://www.mrfizzix.com/archery/bow.html explains it better than I can.

    Because it is 7th grade I'm not going to go into too great of depth, but I want there to be some understanding of the physics involved.


    As I said, this is a work in progress, and I appreciate the feedback.
    cheers!
    chriso
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    Post by phuphuphnik on Mon Feb 24, 2014 1:53 am

    I also just thought of this. The force required to draw and hold the string is not the force put onto the arrow. It is what is required to hold it. The force is a function of the spring being bent, and the change in the angle of the spring. The inertia of the limb issue is solved by using the draw length in the second equation. This explains why a slower steel bow with a long draw is similar to a fast fiberglass one with a short pull. Energy is always conserved. The energy used to get the heavier bow moving is resolved by imparting the energy over a longer distance.Kind of...
    cheers
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    Post by Hotspur on Tue Mar 04, 2014 11:27 pm

    Have you considered a lead projectile? Around 600 gn weight?
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    Post by phuphuphnik on Wed Mar 05, 2014 9:20 am

    Lead would be ideal, but we're going to use them on school grounds. The level of little kids and worrying parents make lead a bad idea. I was thinking taconite (iron ore) pellets.
    Mine does great with .495 lead roundballs.
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    Post by Hotspur on Fri Mar 07, 2014 7:07 pm

    The discussion of the qualities of material for the projectile sounds like an excellent physics class.

    Particularly as you will have the ability of putting theory to the test.

    I wish I had had such a creative teacher as yourself in grade school!
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    Post by Jack Pine on Sat Mar 08, 2014 1:33 am

    phuphuphnik wrote:Lead would be ideal, but we're going to use them on school grounds. The level of little kids and worrying parents make lead a bad idea. I was thinking taconite (iron ore) pellets.
    Mine does great with .495 lead roundballs.
    Taconite, now there's a work I have neither heard nor used since my days as a lad in MI's UP. I used to walk along the railroad tracks and pick up all the taconite pellets [balls] I ever wanted. Made great ammo for my sling shot.

    Jack ><>

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