(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.
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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.
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.