How to compute FPE?

Wow.. Gravity got updated since I learned the formula way back in the 1900's. Talk about feeling old, but hey learning isn't something you supposed to stop doing anyway. The formula still is the same just changing the divisor to reflect the new more precise gravity derivitive. Honestly just to make remembering the formula easier, a wee bit of rounding can be used, changing the divisor from 450436 to 450450 yeilds the same accuracy out to 3 decimal places.

So my newly revised (easier to remember) formula submission:

{(weight in grains)*(fps)*(fps)}/450450


 
the general forumla for kinetic energy is not all that is being calculated, so yes the conversion is needed in the calculation. 
https://en.wikipedia.org/wiki/Muzzle_energy
and since most people weigh things by measuring the force of gravity on the mass of an item and since we implicitly understand gravity to be a constant, we can say the force of gravity on the item (or the weight) equals the mass, so yes, it would be weight in grains. if we were measuring with an inertial mass measuring device then we would use mass.
 
Like I said...

Gravity is not needed. Some early experimenters probably did not know how to convert grains(mass) to slugs(mass). They may have been more familiar with pounds. So they converted grains to lbf (pounds force) rather than lbm (pound mass). At which point they realized they needed gravity to get the units to jive. That is where the error came in. They used a roundabout way via gravity. With all it's pitfalls.

Your muzzle energy will be the same, even in the absence of gravity (deep space). So, no. Gravity is not needed to determine muzzle energy

Also: most all scales are calibrated to a known mass. So even though some scale may be using the resulting force of gravity as a measurement, they are calibrated to readout mass. The exception would be something like a trigger "weight" scale, which is calibrated to readout force.

A scale, such as a balance beam scale, always compares the mass being weighed to an existing mass. So the value of local gravity is irrelevant. That type of scale reads the same on the moon or on earth. It does not matter what value of gravity is used.

A spring scale measures a deflection distance to determine a force. If that force is the results of a mass under acceleration, and we know the acceleration (local gravity is a convenient means of acceleration), we can determine the mass. The value for the local acceleration of gravity is already built into a calibrated scale.

The conversion of grains to slugs:
https://www.google.com/search?q=grains+slugs&ie=utf-8&oe=utf-8#q=slug+grain

That's where the 450436 comes from. (2 x 225218).

You can use gravity in the equation if you really want to. Just make sure you are using the correct value. But it's not needed. So why complicate things?
 
Adam,
I read your source material.

They converted grains to lbs, and used the value of gravity to convert lbs to slugs. It's fairly simple (at least until you insert gravity into the equation). Easier to convert directly to slugs. Though most are not comfortable using slugs so they do the intermediate step of converting to lbs. That's where the discrepancy for calculating FPE came into being. They used a slightly wrong conversion, but since the final answer is still close enough, it stuck.

7000gr/lbm x 32.174lbm/slug = 250218gr/slug (that is correct)

7000gr/lbm x 32.163lbm/slug = 250141gr/slug (that is NOT correct)

https://www.google.com/search?q=slugs+pounds&ie=utf-8&oe=utf-8#q=slugs+pounds+conversion

https://www.google.com/search?q=slugs+pounds&ie=utf-8&oe=utf-8#q=pounds+grains+conversion

https://www.google.com/search?q=slugs+pounds&ie=utf-8&oe=utf-8#q=slugs+grains+conversion