question about arrow speed math,,?

[bradsmith2010]

if you have a 50# bow that will shoot 185fps with 7.5 gpp,,,(375grain arrow)
how fast will it shoot with a 500 grain arrow(10ggp) ,, is there an easy way to calculate that?? 

[loon]

it depends on how much energy it stores? If it's an efficient bow that stores little energy, then it'll be much slower, if the efficiency difference would be less. Or if it's an inefficient bow that stores lots of energy, it'd only be a little slower in comparison, the efficiency difference would be greater?.

you can calculate the minimum speed for the 500grain arrow assuming efficiency is the same (for momentum, I guess) with both arrow weights (it'll be higher with the heavier arrow).

momentum = mass*velocity
375grains*185fps = 69375  ..
500grains*x = 69375
69375/500 = 138.75 fps

Now if we assume kinetic energy is constant

KE=m*v^2 / 2
375grains*(185fps)^2 / 2 = 12834375 / 2
500*x^2 / 2 = 12834375 / 2
sqrt(12834375/500) = 160.21fps

..
Chances are, both kinetic energy and momentum would be greater with the heavier arrow, but momentum would increase more. so it'll probably be at least 160fps? I guess...

[bradsmith2010]

thank you Loon, that gets me in the ballpark and sounds reasonable,,  I have a bow that is shooting about 160fps 10gpp,, so I will try it with a 375 grain arrow  with the chrono and see what it does,,

[Danzn Bar]

Woo Weee my head hurts ...
DBar

[bradsmith2010]

yes I was hoping I could learn how to calculate that,, no way, I will have to do it in real time trial and error with a bow arrow and chrono,,

[loon]

Just from what I know of physics, but I'm not sure of what'll happen, so I'd like to know what speeds you get with a 500 grain arrow..

I guess kinetic energy doesn't decrease until arrows are extremely heavy (2000+ grains? Dunno) so calculating with constant KE could be more useful than with constant momentum. It's just because..

speed * mass = momentum
(1/2) * speed * speed * mass = kinetic energy

I have no idea why this is the case or what they really represent, other than some stuff about work being force times distance (the force of the bow's pull and the distance of the draw length) and sort of corresponding to kinetic energy as potential energy...

(1/2) * 185fps * 185fps * 375grain = 6417187.5 fps^2 grains = 38.6 Joules according to Wolfram Alpha...

So now, it's just Algebra. We take a 500 grain arrow. The speed s is unknown, what we want to find. Realistically, KE would increase some amount with the heavier arrow but in this calculation I assume it's the same (6417187.5), so this would be a minimum expected arrow speed; it'd probably be more.

(1/2) * s * s * 500grain = 6417187.5 fps^2 grains
*2 both sides

s * s * 500grain = 12834375 fps^2 grains
/500grains both sides

s * s = 25668.75 fps^2
square root both sides

s = 160.21fps

Assuming the KE was constant, the momentum would now be

500grains * 160.21fps = 80105.00grains*fps = 1.58 Newton*seconds

Before, it was

375grains * 185fps = 69375grains*fps = 1.37 Newton*seconds. 86.71% as much momentum.

I wish physics was still this fun in university :\ But I just sort of guessed that.. KE would increase more slowly than momentum as arrow weight increases. It makes sense but now I don't even know exactly why.

[Del the cat]

No! There is no easy way to calculate it, in fact there is no way to calculate it at all!
It can be estimated, but probably no more accurately than you could guess.
Next!
You have to remember, maths is just a way of trying to explain the world around us. There is no way even the most sophisticated computer in the world can accurately model what is happening in a self wood bow.
If you enjoy messing with maths and physics, fine, but it is just a tool.
Otherwise shoot it through a chrono with different weight arrows and plot a graph of the results, of just chill and have a beverage of choice.
Del
(cynical old git mode )

[crooketarrow]

  I've never did the math. But I know just from building bows. And chono all my bows for the 15 years.

  A well built and tillered 50# bow shooting a matching spine 500 grain arrow should shoot. Will shoot 160 FPS and 170 FPS reachable.

  WOOD IS WOOD why some bows built the same. Out shoot the others. WOOD IS WOOD.

[Badger]

  Loon, you seem pretty proficient at math. You might be interested in learning how to use virtual mass. Virtual mass can be used to isolate hysterisis from your regular energy losses. Virtual mass is kind of a pain to calculate but it can be very useful to work with. If you can measure an accurate measurement for stored energy that will tell you how fast a bow could shoot an arrow if it were 100% efficient. Say you shoot a 500 grain arrow at 170 fps. Now you figure out how much arrow weight you could shoot at 170 fps using all the bows energy. It might be something like 750 grains. This would give you a virtual mass of 250 grains. With a bow with no hysterisis this figure will remain fairly constant regardless of arrow weight. Your efficiency will go down as arrow weight goes down but your virtual mass will remain fairly constant. When it comes to wood bows your virtual mass will continue to climb as arrow weights go down due to hysterisis which is very sensitive to speed. If you bow does maintain a constant virtual mass you have done a most excellent job tillering and hysteris is very low.

[bradsmith2010]

ok I did a test with a bow about the same
47#26 inches shooting a 350 grain arrow 178 fps  7.44 GPP
I also shot it with 472 grain arrow            162fps   10.02 GPP

please keep in mind my release is average and I was just pulling to a piece of tape with no spotter

Loon your 160 fps seems pretty reasonable,, do you think this would be relative say for a 60#  bow , of same design  shooting the same GPP

[BowEd]

Good thread here brad.Your bow is shooting 15 fps above average then correct?Pretty good for only pulling to 26".
The bow has to be shot that's for sure in my book to see how efficient it is.So in your case does it mean that your bow would shoot a 415 grain arrow 170 fps?If so then each 61 grains added to your arrow weight your bow will shoot 8 fps slower correct?Or a 533 grain arrow 154 fps.Or a 594 grain arrow 146 fps.I can make these assumptions because of what the bow has shown you.
If I'm correct here your bow is shooting a 10gpp arrow with the speed of an average 62# bow.Very good.That to me is what it's all about.

[BowEd]

The efficiency of the bow is not known till you shoot it.

[BowEd]

Badgers' comments align with Karpos' Turkish horn bow book that your ratio of mass to draw weight if kept low will produce a faster shooting bow.

[BowEd]

I wonder what the average was again about grain weight on arrows to loss in fps.Seems like I read it somewhere but can't remember it.

[Badger]

  On a 50# bow shooting a 10 grain per pound arrow I figure I will gain about 14 fps with 100 grains less weight. If I go from 250 grains to 150 grains I pick up about 60 fps roughly. If you add 100 grains to 10 grains per pound it might go down only about 10 fps. The virtual mass of each bow gives the best way of estimating arrow speeds with different weight arrows, on wood bows it is not very accurate though.

  The best way to grasp it is to think of it like this. If you determine your bow has a virtual mass of 200 and you shoot a 200 grain arrow the most efficient you can possibly be is 50% because half went to the bow. If you are shooting a 400 grain arrow 66 2/3 efficient would be the best possible because your arrow weight is 2/3 and your virtual mass is 1/3.