The Firing Line Forums

Go Back   The Firing Line Forums > The Conference Center > General Discussion Forum

Reply
 
Thread Tools Search this Thread
Old November 7, 2017, 04:50 PM   #1
JohnKSa
Staff
 
Join Date: February 12, 2001
Location: DFW Area
Posts: 24,932
Simple formula for calculating kinetic energy...

(grains/100) x (fps/100) x (fps/100) x 2.22 ≈ ftlbs

The error in the approximation is less than 0.004%--negligible for firearm energy calculations.

I came up with this approximation awhile back and thought others might find it handy. If you want to know how/why it works, keep reading.
***************************************************
The formula for kinetic energy is:

1/2 m v^2

It's common to see energy calculated using the following formula:

mass x velocity x velocity ÷ 450436 = ftlbs

The divisor is not really a whole number, it's actually something like:

450436.6797900260...

That number is the product of 2 (from the 1/2 in the 1/2 mv^2 formula), 7,000 (the number of grains in a pound) and 32.1740485564304... which is the number of pounds in a slug at standard gravity.

It is necessary to convert from grains to slugs because the energy formula calls for mass and the unit of mass in the feet-pounds system is the slug.

The precise value of the divisor isn't really important since it turns out that any number that looks like 450xxx is a pretty good approximation. In fact, that's how this particular approximation works.

This approximation is based on the fact that 1/222 = 0.00450450450450450...

Where is this going? Well, if you multiply that number by 100 million you get 450450.45045045... A number that is close to the actual divisor value. Very close--the error is less than 0.0031%

An error that small means that for typical energy figures, using the approximation will have a negligible effect on the result.

For example, if the actual energy figure is 350ftlbs, the error in the result calculated with the 1/222 x (100,000,000) would be about 0.011 ftlbs. Since energy figures are commonly rounded to the nearest whole number, an error that small is meaningless. Even when calculating very large energy figures--say for a .50BMG, the approximation error is only about a third of a foot pound.

222 is obviously a lot easier to remember than 450436, but keeping track of how many zeroes to poke into the calculator when you're multiplying by 100 million is kind of a pain. Fortunately there's an easier way to handle it.

100 x 100 x 100 x 100 is 100 million. So we shift the decimal point in 222 by two places to get 2.22 (there's one of the four 100s we need) and then we shift the decimal point in the grains and fps values by 2 places to get the other three 100s we need. Remember velocity is used in the formula twice, so dividing it by 100 means dividing the final result by 100 x 100.

Divide both the grains and velocities (in fps) by 100 before starting the calculation. Easily done by just shifting the decimal points to the left by 2 places in each number. Then multiply by 2.22

(grains/100) x (mass/100) x (mass/100) x 2.22

For a 125 grain bullet at 1450fps, that would look like the following:

1.25 x 14.5 x 14.5 x 2.22 = 583.44375

Using the actual formula gives the following result:

125 x 1450 x 1450 ÷ 450436.6797900260... = 583.461586926073...

Rounding both of the results to the nearest whole number gives 583 ftlbs for each one.
__________________
Do you know about the TEXAS State Rifle Association?
JohnKSa is offline  
Old November 7, 2017, 06:28 PM   #2
ShootistPRS
Senior Member
 
Join Date: January 3, 2017
Posts: 1,583
Your short hand version may get close with some velocities and some bullet weights but it's off by just over 32 ft.lbs. for a 3000 fps 180 grain bullet.
The actual ME is 3596.5 and your calculation calls it 3564; that is 32.5 ft lbs difference or 0.9% off. I recommend that you try different bullet weights and velocities from high to low and heavy to light to further refine your method. The 450436 number is the number of grains in a pound times 2G or 7000 times 64.348. G=32.174 fs^2.

It's a good first attempt but you need to expand it a bit. (for old codgers like me)
ShootistPRS is offline  
Old November 7, 2017, 10:30 PM   #3
JohnKSa
Staff
 
Join Date: February 12, 2001
Location: DFW Area
Posts: 24,932
Quote:
The actual ME is 3596.5 ...
Correct.
Quote:
...your calculation calls it 3564...
180grains and 3000fps, using the approximation, would be:

1.8 x 30 x 30 x 2.22 = 3596.4

That's a magnitude error of about a tenth of a foot pound.

The approximation will be within 0.0031% of the correct value, no matter what the values for grains and fps are. For any typical firearm calculations the approximation values will be correct to within a fraction of a ftlb.
Quote:
The 450436 number is the number of grains in a pound times 2G or 7000 times 64.348. G=32.174 fs^2.
Correct.

"That number is the product of 2 (from the 1/2 in the 1/2 mv^2 formula), 7,000 (the number of grains in a pound) and 32.1740485564304... which is the number of pounds in a slug at standard gravity."
Quote:
I recommend that you try different bullet weights and velocities from high to low and heavy to light to further refine your method.
While the raw error will grow with the magnitude of the energy value, it is still going to remain quite small for any energy values one would expect to encounter for typical firearms. And, of course, the percent error remains constant.

Below is a chart comparing the calculation and the approximation for a variety of bullet weights and velocities from very small to very large. Until the energy level gets above 16,300 ftlbs or so, the raw error will remain below half a ftlb.

grains----fps----KE (act.)--KE (approx.)--Error (ftlbs)--Error (%)
30-------400-----10.66--------10.66-------0.0003------0.00306%
50-------750-----62.44--------62.44-------0.002-------0.00306%
100----1000----222.01-------222.00-------0.007------0.00306%
200----2000---1776.05-----1776.00-------0.054------0.00306%
300----3000---5994.18-----5994.00-------0.183------0.00306%
400----3000---7992.24-----7992.00-------0.244------0.00306%
500----3000---9990.31-----9990.00-------0.305------0.00306%
600----3000--11988.37---11988.00-------0.366------0.00306%
700----3000--13986.43---13986.00-------0.428------0.00306%
750----3500--20396.87---20396.25-------0.624------0.00306%
__________________
Do you know about the TEXAS State Rifle Association?
JohnKSa is offline  
Reply

Thread Tools Search this Thread
Search this Thread:

Advanced Search

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off

Forum Jump


All times are GMT -5. The time now is 11:02 PM.


Powered by vBulletin® Version 3.8.7
Copyright ©2000 - 2024, vBulletin Solutions, Inc.
This site and contents, including all posts, Copyright © 1998-2021 S.W.A.T. Magazine
Copyright Complaints: Please direct DMCA Takedown Notices to the registered agent: thefiringline.com
Page generated in 0.05702 seconds with 8 queries