Arrow shot from Bow
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I wonder whether anyone could help me recall something. I remember years ago I found a simple proof ... or rather mathematical demonstration that the optimum length for an arrow to be drawn back to be shot from a bow is just about equal to the length of the arrow: that in the course of analysis of stability under small departure of the arrow from orthoplicity in its initial placing in the bow at the moment of shooting, and of efficiency of transfer of energy from the bow to the arrow, etc, basically all the particulars - the stiffness of the bow, the length & mass of the arrow, all cancel & drop out, & leave as a 'residue' (shall we say?) that the optimum shooting position is the arrow drawn-back such that its tip is prettymuch at the body of the bow.
This is one of those elementary items of knowledge that we carry around with us, not so much in or conscious reasoning minds as in our musculature & its innervation & encoded into or faculty of propriorecetion (immediate awareness of the the configuration & state-of-action of our musculature). We just know without thinking about it that that is the optimum distance for an arrow to be drawn back; but I remember that it can be shown to be by a surprisingly simple (but nevertheless rather cunning) analyisis of the matter, importing dimensionality & stabilty, & applying these to the differential equation of the arrow's motion in the course of its being shot, without explicitly solving those equations.
So I'm wondering whether anyone can guide me in the recovering of this analysis from my memory.
differential-equations stability-theory
asked Nov 24 at 9:11
AmbretteOrrisey
45810
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-1
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I wonder whether anyone could help me recall something. I remember years ago I found a simple proof ... or rather mathematical demonstration that the optimum length for an arrow to be drawn back to be shot from a bow is just about equal to the length of the arrow: that in the course of analysis of stability under small departure of the arrow from orthoplicity in its initial placing in the bow at the moment of shooting, and of efficiency of transfer of energy from the bow to the arrow, etc, basically all the particulars - the stiffness of the bow, the length & mass of the arrow, all cancel & drop out, & leave as a 'residue' (shall we say?) that the optimum shooting position is the arrow drawn-back such that its tip is prettymuch at the body of the bow.
This is one of those elementary items of knowledge that we carry around with us, not so much in or conscious reasoning minds as in our musculature & its innervation & encoded into or faculty of propriorecetion (immediate awareness of the the configuration & state-of-action of our musculature). We just know without thinking about it that that is the optimum distance for an arrow to be drawn back; but I remember that it can be shown to be by a surprisingly simple (but nevertheless rather cunning) analyisis of the matter, importing dimensionality & stabilty, & applying these to the differential equation of the arrow's motion in the course of its being shot, without explicitly solving those equations.
So I'm wondering whether anyone can guide me in the recovering of this analysis from my memory.
differential-equations stability-theory
asked Nov 24 at 9:11
AmbretteOrrisey
45810
1
Recommend moving to physics SE
– Eddy
Nov 24 at 12:53
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I wonder whether anyone could help me recall something. I remember years ago I found a simple proof ... or rather mathematical demonstration that the optimum length for an arrow to be drawn back to be shot from a bow is just about equal to the length of the arrow: that in the course of analysis of stability under small departure of the arrow from orthoplicity in its initial placing in the bow at the moment of shooting, and of efficiency of transfer of energy from the bow to the arrow, etc, basically all the particulars - the stiffness of the bow, the length & mass of the arrow, all cancel & drop out, & leave as a 'residue' (shall we say?) that the optimum shooting position is the arrow drawn-back such that its tip is prettymuch at the body of the bow.
This is one of those elementary items of knowledge that we carry around with us, not so much in or conscious reasoning minds as in our musculature & its innervation & encoded into or faculty of propriorecetion (immediate awareness of the the configuration & state-of-action of our musculature). We just know without thinking about it that that is the optimum distance for an arrow to be drawn back; but I remember that it can be shown to be by a surprisingly simple (but nevertheless rather cunning) analyisis of the matter, importing dimensionality & stabilty, & applying these to the differential equation of the arrow's motion in the course of its being shot, without explicitly solving those equations.
So I'm wondering whether anyone can guide me in the recovering of this analysis from my memory.
differential-equations stability-theory
asked Nov 24 at 9:11
AmbretteOrrisey
45810
I wonder whether anyone could help me recall something. I remember years ago I found a simple proof ... or rather mathematical demonstration that the optimum length for an arrow to be drawn back to be shot from a bow is just about equal to the length of the arrow: that in the course of analysis of stability under small departure of the arrow from orthoplicity in its initial placing in the bow at the moment of shooting, and of efficiency of transfer of energy from the bow to the arrow, etc, basically all the particulars - the stiffness of the bow, the length & mass of the arrow, all cancel & drop out, & leave as a 'residue' (shall we say?) that the optimum shooting position is the arrow drawn-back such that its tip is prettymuch at the body of the bow.
This is one of those elementary items of knowledge that we carry around with us, not so much in or conscious reasoning minds as in our musculature & its innervation & encoded into or faculty of propriorecetion (immediate awareness of the the configuration & state-of-action of our musculature). We just know without thinking about it that that is the optimum distance for an arrow to be drawn back; but I remember that it can be shown to be by a surprisingly simple (but nevertheless rather cunning) analyisis of the matter, importing dimensionality & stabilty, & applying these to the differential equation of the arrow's motion in the course of its being shot, without explicitly solving those equations.
So I'm wondering whether anyone can guide me in the recovering of this analysis from my memory.
differential-equations stability-theory
differential-equations stability-theory
asked Nov 24 at 9:11
AmbretteOrrisey
45810
asked Nov 24 at 9:11
AmbretteOrrisey
45810
edited Nov 24 at 15:58
asked Nov 24 at 9:11
AmbretteOrrisey
45810
asked Nov 24 at 9:11
AmbretteOrrisey
45810
asked Nov 24 at 9:11
AmbretteOrrisey
45810
45810
1
Recommend moving to physics SE
– Eddy
Nov 24 at 12:53
add a comment |
1
Recommend moving to physics SE
– Eddy
Nov 24 at 12:53
1
1
Recommend moving to physics SE
– Eddy
Nov 24 at 12:53
Recommend moving to physics SE
– Eddy
Nov 24 at 12:53
add a comment |
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Recommend moving to physics SE
– Eddy
Nov 24 at 12:53