# Calculating muzzle speed from bullet drop without a chronograph

airscopes
Forgive my naivety. I used to know math, but have sort-a fallen off the bicycle some 30 years ago. I hope I don't go all ballistically ignorant.

I have a problem trying to calculate muzzle velocity for pellet rifles. Specifically, I assume there is a way to do it without a chronograph and come out more accurate than current solutions and without the complexity (the ballistic pendulum: http://hyperphysics.phy-astr.gsu.edu/hbase/balpen.html ; MV calculator: http://www.airgunexpo.com/calc/calc_mve.cfm? ). The online calculator assumes you are using sights or a scope. I guess I feel that any time you add a scope into the equation (pretty literally here) you are just adding variables that there are no real way to account for. The ballistic pendulum seems helpful, but may be a bit difficult to set up correctly so the data will be useful.

A bore sighter points a laser out of the rifle bore for the purposes of alignment. I would think there is a way to use the sighter to get an accurate drop at a known distance, and figure out from the fall how fast the pellet was launched. With a bore sighter and a 0 degree angle of launch, it seems you should be able to simplify the calculation AND use one target (the calculator requires two -- and I believe that is another source of error).

The angle of launch is never mentioned in the calculator... That could goof things up significantly (e.g., if you shoot vertically into two targets at 90 degrees, they could have infinite speed, which is obviously not correct).

The process of obtaining data would be:

2. Cock
3. Level barrel using a bench rest or gun vice to 0 degrees
4. Insert bore sight (should be on)
5. Align target. The target should be positioned exactly a certain distance from the end of the barrel. Easiest to mark the target where the laser is pointing by dotting the laser spot by adding a sticky-back target dot.
6. Remove bore sight
7. Shoot
8. Measure drop on target between the point of impact and target dot

You will know the distance between the target and the front of the barrel. Convert it to feet. Call it D.
You will know the distance the pellet dropped due to gravity. Call it H.
As gravity works as a constant (32.15 ft/sec^2). Call it G.
You want to know the velocity. Call it V.

I think you can use H = 1/2 * G * T^2
We don't have T, but T = D/V, which gives you: H = 1/2 * G * (D/V)^2
Solving for V... V = D * (G / 2H) ^ 1/2

So say you were shooting at 100 yards. You have to convert to feet... 300 feet
We know G is 32.15
If the pellet dropped 1.5 inches...

V = 300 * (32.15 / 2 * 1.5) ^ 1/2
V = 982 fps

The variation is about 3.5 inches for 600 to 1000 fps at 100 yards. This a good mid-high range for pellet gun velocity.

At 25 yards, measuring becomes even more critical... and therefore with more error.

So say you were shooting at 25 yards. You have to convert to feet... 75 feet
We know G is 32.15
If the pellet dropped .25 inches...

V = 75 * (32.15 / 2 * .25) ^ 1/2
V = 601 fps

You might see short distances are impractical for measuring. A 1000 fps pellet should only drop about .09 inches. That means the difference between 600 fps and 1000 fps is a drop of only .15 inches... it is getting harder to measure, and harder to determine if the result is mostly error in measuring.

The calculations above work in a vacuum, and probably none of us will be building a 100 yard long vacuum in our back yard as the purpose here is to get it done on the cheap. And of course there will be MORE error at longer distances because of the drag. SO... in a vacuum we have error at short distance, and in the real world there may be as much or more at long distances...and some at both.

But accounting for drag may offer a better mid-range shooting opportunity. That is, if the bullet drops much more because of drag and we can shorten the distance of the testing to 50, 40 or even 30 yards because drop is greater and easier to measure than it would be in a vacuum. So, there may be a good compromise of distance and drop range. But, I'm a little puzzled how to work the BC (ballistic coefficient) into this right now... Luckily, I have a ballistic coefficient chart that covers a wide range of pellets:

http://www.photosbykev.com/wordpress/userfiles/pelletdata.htm

The problem becomes what to do with the BC. I've read a few things, but don't really see calculations that go much beyond figuring what the BC is... I have it and want to know how to use it. I assume from the BC that I can adjust the calculation done above to include the influence of drag over the selected distance... My math-mind stops right there.

Any help?

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V = 300 * (32.15 / 2 * 1.5) ^ 1/2

I haven't made my way past this point in your post, but here you are mixing feet and inches. You need to use H= 1.5/12 = 0.125 feet in the calc (or else use G = 32.15*12 = 386 inch/sec2).

airscopes
hahaha... oops. thanks. that changes the calculations quite a bit!

So say you were shooting at 100 yards. You have to convert to feet... 300 feet
We know G is 32.15
If the pellet dropped 1.5 feet

V = 300 * (32.15 / 2 * 1.5) ^ 1/2
V = 982 fps

The variation is about 3.5 feet for 600 to 1000 fps at 100 yards. This a good mid-high range for pellet gun velocity.

At 25 yards, measuring becomes even more critical... and therefore with more error.

So say you were shooting at 25 yards. You have to convert to feet... 75 feet
We know G is 32.15
If the pellet dropped 3 inches... or .25 feet

V = 75 * (32.15 / 2 * .25) ^ 1/2
V = 601 fps

That is about 2 inches between 600 and 1000 fps @ 25 yards! it changes the problem significantly to defining how the BC works.

Thanks for noting that rather glaring error!

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Yeah, but is the new result (3400 ft/sec) realistic? I don't know but it seems high (I seem to recall a .45 ACP pistol is about 1000 ft/sec, I don't have a feeling for rifle velocities).

If the result is too high what does that mean?

airscopes
it should be the drop in feet, not inches... so it is 1.5 feet / 18 inches! See the redone calculations.

3400 fps would be way high for a springer pellet rifle.

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If I understand the definition of the BC factor, it is a percent decrease in velocity per yard traveled.

That means that the velocity is decreasing exponentially with distance traveled from the muzzle. You can approximate this in a spreadsheet (look at velocity at 0,1,2...100 yards) or you can use velocity = (muzzle velocity)*exp(-BC*x) (just keep track of the percent (divide BC by 100) and do x in yards.

Does that make sense? If not I can try to write up more later (back to work)

Your links provide values for the ballistic constant BC with units % per yard. I'm going to work in feet and seconds, so I define

B = BC / 100 / 3

B is then related to velocity V by

$$\frac{dV}{dx} = -BV$$

solve for V

$$V = V_{0} e^{-Bx}$$

where Vo is the initial velocity when x = 0.

We also need velocity V as a function of time. We had

$$\frac{dV}{dx} = -BV$$

multiply by dx/dt on both sides and simplify, recognizing that dx/dt is equal to V (don't show the mathematicians)

$$\frac{dV}{dx} \frac{dx}{dt} = -BV \frac{dx}{dt}$$

$$\frac{dV}{dt} = -BV^2$$

Solving for V(t), and rearranging to get t:

$$t=(\frac{1}{V} - \frac{1}{V_{0}})*\frac{1}{B}$$

finally you can calculate H, the bullet's 'drop,' from t

$$H = -\frac{1}{2} g t^2$$

So for example if BC is 0.35%/yard (giving B=0.001167) and Vo is 1000 ft/sec, we can calculate at x=300 feet:

$$V = V_{0} e^{-Bx}$$
$$V = 1000 e^{-0.001167*300} = 704.7 ft/sec$$

and then
$$t=(\frac{1}{V} - \frac{1}{V_{0}})*\frac{1}{B}$$
$$t=(\frac{1}{704.7} - \frac{1}{1000})*\frac{1}{0.001167} = 0.3592 seconds$$

and the drop is
$$H = -\frac{1}{2} g t^2$$
$$H = -\frac{1}{2} 32.2 (0.3592)^2 = 2.08 feet$$

Does that make sense?

airscopes
I'm looking it over and I think I need to look it over more. Thanks for taking the time... I may have another question or few...

This seems to go backward. I want to determine initial velocity with distance, drop, gravity, and BC... because I have to assume the initial velocity is unknown. But this at least seems to give me a hint as to where to go. I just need to digest the meal.

thanks!