Phil,
Energy = 1/2mv^2, or in words, energy equals one-half the mass of the bullet times the square of the velocity (velocity times itself). [If I am being too simplistic, I apologize, but there might be some other readers who don't have science or engineering backgrounds.]

To get the figures right, some fudge-factors have to be thrown in since we tend to use some different measurement systems that complicate things. There are, for example 7000 grains to a pound. But a pound is a unit of WEIGHT, which depends on the acceleration due to gravity, which, at sealevel is about 32.17 ft/sec/sec or ft/sec^2. Then we've got that pesky 1/2 in the equation. So multiplying and dividing we get a figure of 450380, which is thrown into a simplified equation.

Thus, a 250 gr bullet (dia. has NO bearing on this calculation) that has a velocity of 900 ft/sec (feet per second) has an energy of 250 x 900 x 900) divided by 450380 = 450 ft-lbs...or foot-pounds (and that is correct, it is NOT ft/lbs!)

"Muzzle" energy is simply the calculated value obtained from muzzle velocity. As was explained by others, we seldom take the velocity at the muzzle since the bullet must pass through the screens of a chronograph, and so we really determine the velocity at some distance, 7 ft, 10 ft, or whatever, depending on the setup. This may or may not be corrected using complex computations, to give a "true" muzzle velocity.

Question is, who cares? At the velocities most of us shoot, there isn't going to be a big difference between "instrumental" and "muzzle" velocities. Even when hunting, the differences are miniscule.

Is muzzle energy the true indicator of so-called "knockdown" power? To start with, unless you are in the habit of running up to a deer or bear, sticking the muzzle of the gun (3-1/2" barreled sheriff's model??? GGG!), the muzzle figures are meaningless. This is because the bullet starts to slow down the instant it leaves the barrel, due to air resistance. But it does give us a starting point! Shape, velocity (sub-sonic, transsonic and supersonic velocity changes the way air acts on the bullet), and mass per cross sectional area, all affect the rapidity with which the bullet falls off after exitiing the barrel.

Remember, the energy changes as the square of the velocity. So if the 900 ft/sec bullet slows to 750 ft/sec, the energy drops from 450 ft-lbs to 312 ft-lbs. The velocity falls 17 percent. But the energy drops by 31 percent! Nearly double the loss.

That's fine, but it is NOT an indication of how the bullet will perform in terms of knocking down a steel plate target, OR game.

Knocking down a steel plate target basically depends on the mass of the target, the mass and velocity of the bullet, and whether the bullet holds together long enough to impart all or most of its energy to the target. (The momentum calculation...simply MASS times VELOCITY, and that may affect whether the target is knocked over, along with the friction in the hinges, if any!)

Different, and VERY complicated deal when talking about "knockdown" power on game...or TLS (two-legged skunk). Bullet velocity (hence, energy) DOES have a bearing on KDP, but size, shape of the bullet as it strikes the target, bullet expansion or lack thereof (see Evans and Sanow); whether the bullet is sub- or supersonic on impact; stability of the bullet on striking; whether only soft tissue is struck or bone; whether secondary fragments cause damage to blood vessels and vital tissues; whether the central nervous system is affected, and how much, are some of the factors that determine "knock-down" power. Some of this IS dependent on bullet velocity AT THE POINT OF IMPACT.

If it sounds complicated, it IS! Books have been written. Everything from ballistic gellatin (super-Jello), to goats, pigs, cows have been experimented with over the last century or so, and in some instances, post-mortum studies, conducted. AND THERE STILL ARE NOT SO MANY DEFINITIVE ANSWERS. That is why ammunition companies spend big bucks determining a better way to build a bullet.

At the final analysis, the old Mountain Man's adage still holds true on game..."Ya gotta tech the life!" Hope that helps a bit, pard!