Trajectory

I’ve mentioned before that I’m trying to round out the absolute fundamentals this month.  One thing I’ve totally neglected thus far is an in-depth discussion of sighting and aiming.  I have given it a lot of thought and I just don’t feel like doing it.  My choice of what to write has been biased towards what I’m doing with the equipment I have.  Since I use a scope, I don’t think it warrants an entire article to say, “Put your eye in the spot where you get a full edge-to-edge picture through your ocular lens and put the crosshairs on the target.”

I have the most intelligent readership in the United States.  How do I know this?  My daily readership accounts for 0.00000018% of the US population.  I can safely deduce that the people who want to read what I write are the smartest in any population.  We, therefore, represent the absolute tip of the spear in terms of intelligence and discernment.  Therefore I have elected to dispense with telling you to align your sights, get a good sight picture and focus on your front sight.  If you can’t handle that, email me and I’ll put the second half of the preceding sentence as a standalone article entitled “Using Your Sights”.

In place of an article on sighting and aiming, we’ll cover trajectory.  You’ll get a lot more out of it anyway.  Read on, if you think you can handle it.

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Trajectory can be thought of as akin to voodoo for the budding rifleman.  We’re aware on some level that the bullet doesn’t exactly hit at the same point of aim/point of impact at 600 yards as it does at 100 yards.  What seems like some sort of magic is that the more experienced rifleman can adjust for that difference.

When it comes down to it, bullet drop is pretty easy to predict with some basic data.  The science is apparently quite “mature” for this type of thing.  So let’s unravel a tiny bit of it here.

After you fire your bullet at your target, the bullet drops as it travels forward.  If you were to align your rifle with the barrel parallel to the ground and fire a round, while at pecisely the same time dropping a bullet from the same height, the two bullets would hit the ground at the same time.  Something we do to cheat this is to fire the bullet at a slight upward angle, so that the time of flight and distance covered by the bullet can be increased.  The angle varies based on how far we want the bullet to go.

 

When we zero our rifle, we adjust the rifle’s sighting system, which is our line of sight, in a way that causes us to angle our barrel, which is the bullet’s line of departure, just enough so that our bullet will intersect with our line of sight at a specific point.  The bullet may, in fact, cross our line of sight twice.  Because our sights are mounted above the bore, the bullet needs to rise to make it to the line of sight.  This first intersection is called the “initial intersection” or “double I” or “II”.  The bullet will likely keep rising slightly, then fall, crossing our line of sight again at our zero distance.  Ironically, the farther the zero distance, the closer the initial intersection will be.  This is because the compensation in the rifle’s sight has to be more drastic to cause the line of departure to point up at a greater angle.

In the case of a bullet, the amount that it drops isn’t much for the first couple hundred yards, but as the atmospheric drag slows it down, it drops more and more as it travels farther.  Because the science of flying projectiles has been worked out so much, it’s easy to figure out how much it will drop.  To do this you need 1.)  Access to a ballistics program  2.)  The ballistic coefficient of your bullet  3.)  Your muzzle velocity.

Let’s use my original D46 load as an example.  I like to use the Berger Bullets Ballistics Program to get me in the ball park with my comeups.  It’s free and it seems to be right on for me so far.  JBM Ballistics also offers free calculations online.  Lapua provides a G7  ballistic coefficient for the 185 grain D46 FMJ match bullet of .254.  I’ve chrono’d the load and have determined that the average muzzle velocity of the round is 2724 fps.  Below is a copy of what the Berger program gave me.  I had to input the data in that you see in bold print.  The program gave me everything else.  You can choose inches, minute of angle (MOA), or milliradians (mils) for the output.  I chose MOA for this article to keep it simple for those of you that are familiar with it.  I prefer mils:

+—————————– Program Inputs ———————————+
|                                                                              |
+—- Bullet Inputs —–+—-Atmosphere Inputs —-+——-Sight Inputs ——+
| Caliber: 0.308 inches         | Temperature:  70 degrees           | Sight Height:1.53 inches |
| Weight: 185 grains             | Pressure: 27.20 inHg                   | Zero Range: 100 yards   |
| G7 BC: 0.254 lb/in^2          | Humidity:  50 %                             | Look Angle:   0 degrees  |
| G7 Form Factor: 1.097       | Density: 0.06776 lb/ft^3 |                          |
| MZL Velocity:  2724 fps    | Wind Speed: 10 mph       |                          |
|                        | Wind Direction: 3 O’clock|                          |
+————————+————————–+————————–+
+—————————– Program Output ———————————+
       Range    Velocity    Energy     Trajectory         TOF          Drift
      (yards)     (fps)     (ft-lb)               (MOA)         (sec)          (MOA)
           0      2724        3048             0.00         0.0000         0.00
         25      2683        2958             -2.62         0.0277        -0.14
        100      2563        2699            0.00         0.1135        -0.57
        200      2408        2383          -1.80         0.2342        -1.18
        300      2259        2096          -4.31         0.3629        -1.82
        400      2115        1837          -7.19         0.5001        -2.51
        500      1976        1603         -10.41         0.6469        -3.24
        600      1841        1393         -13.97         0.8041        -4.02
        700      1712        1203         -17.92         0.9731        -4.86

        800      1586        1033         -22.30         1.1552        -5.77

 

The “Trajectory” column (italicized) lists the amount that the bullet is above or below the line of sight in MOA.  A minus sign indicates that the bullet is below the line of sight.  This means that you would add this much elevation to your sight to zero at that distance.  This is what is known as a “comeup”.  Notice that the comeup for 25 yards is only a quarter minute different from the comeup at 225.  The comeup for 19 yards (not shown) is identical in this case for the 300 yard comeup.  You would want to verify this in real life before trying it for real, but it would very likely work provided your input data was good.

This load, given the data, would lose its supersonic-ness at around 1200 yards.  The more legitimate phrase for this is “trans-sonic barrier”.  The range at which the bullet enters the trans-sonic barrier is generally considered to be its practical effective range, because crossing the trans-sonic barrier can cause the bullet to lose some of its predicable flight qualities.  My comeup at 1200 yards is projected as 45.68.  I would round that to the nearest quarter and make it 45.75.  Dial that in at 1200 on a windless day (yeah right), aim dead on, fire, and 2.0517 seconds later watch for the bullet strike somewhere in the general vicinity of (maybe even on) the target.

What I would do is verify the data by shooting at some of the distances.  Then I would make a card and put it in my data book, and maybe attach a simplified version to my stock for quick reference.  I hope this demystifies the drop aspect a little bit.

 

     Homemade trajectory quick reference (in mils) on the stock.

 

Ballistic Coefficient

One thing I’ve been mentioning that you may not have heard of is “ballistic coefficient” (or BC).  This is simply a number that describes how well the bullet flies through the air.  The higher the BC, the better the bullet flies, while a low BC bullet is more effected by drag.  Higher BC bullets look sleeker, and you can almost eyeball 2 different bullets and predict which one will fly better.  BC is related to sectional density, and usually longer bullets have higher BC’s than shorter ones.  This means that high BC bullets are heavy for their caliber.

BC is determined by using one of several different “standard projectiles” to provide a comparison for your bullet, which is where the number comes from.  The most common standard projectiles used to describe rifle bullets are the G1 and the G7.  The G1 is the older standard, and is the most common.  The G1 standard projectile looks like an artillery projectile.  If you look up your bullet’s BC, and it doesn’t specify which standard it uses, assume it’s the G1.  In the past few years, Bryan Litz, who is Berger’s ballistician, has demonstrated that the G7 standard projectile provides better data for rifle bullets.  Why?  The short answer is that the G7 standard projectile is shaped more like a modern rifle bullet.  A G7 will be a smaller number than a G1 for the same bullet, usually about half.

Why do you need to care about BC?  It makes a difference what kind of performance you can get.  Remember that my load for the 185 D46 has a G7 BC of .254.  Let’s say I rebarreled my 30-06 to 6.5-06 (the 30-06 cartridge necked down to 6.5mm).  For some reason, there are a lot of high BC bullets in 6.5mm.  Let’s say I found a 140 grain 6.5mm bullet with a G7 BC of .320 (that’s really high, but not unheard of for a 6.5).  Let’s also say that I got it going at 3000 fps, which I think would be a hot load but possible.  Remember that my 30-06 load went trans-sonic at about 1200 yards with 528 ft./lbs of energy.  The 6.5 would go trans-sonic at around 1740 yards!!!  A mile is 1760 yards.  Granted it would only have about the energy of a 9mm at the muzzle (390 ft./lbs), but I wouldn’t want to get hit by it.

A possibly more significant benefit for long range shooting is that a higher BC also means that the bullet is less effected by crosswind.  This means that you’re more likely to hit what your aiming at.  Pretty cool, huh?

What BC doesn’t tell you is how well the bullet is actually constructed.  A better made bullet will typically be more accurate.  High BC bullets, however, tend to have a lot of work put into them, so they are typically well made.  The other thing you can’t infer from a bullet’s BC is what it will do when it hits the target.  Terminal ballistics is a whole ‘nuther animal which I may or may not cover in the future.

Now that we can predict how the bullet will drop, how do we deal with it?  There are basically 2 options: holding and dialing.  Holding means that you’re simply placing your sights on a position other than on the target with the assumption that your point of aim (POA) no longer corresponds with your point of impact (POI).  In terms of bullet drop, the common term for holding off is “Tennessee Elevation”, who is the illegitimate half- brother to the fellow we all know as “Kentucky Windage”.  Dialing means that you’re adjusting your sight mechanism to effectively change your zero so that you can still aim POA/POI.  Holding is generally faster, dialing is generally more precise.  I’ll discuss the how-to’s of holding and dialing in a future article.  I’m already at 4 pages single spaced, which is already too much for one article.

Point Blank Range

One final thing I want to hit on is the concept of “point blank range” (I know you don’t have a problem thinking of PBR as something other than Pabst Blue Ribbon).  Point blank range in the common parlance usually means “really close”.  This is of course wrong, as the 99.99999982% of the people in the US who don’t read my blog usually are.  What PBR really means the maximum range that you can aim POA/POI and still be able to hit a given target.  PBR is determined by a few different things, namely the size of your target, and your bullet’s trajectory.

Here’s how you figure out PBR:  figure out how large your target is. Chuck Hawks tells us that the “vital zone circle” for a medium sized deer is 10″-11″.  Let’s be extra conservative and call it 8″.  Plug your info into one of the ballistics programs.  Set the output as inches.  Check out the results.  Since we’re talking about an 8″ circle, we don’t want to be more than 4″ high or low.  That means you can’t just set your zero at 600 yards and expect to hit everything up to 600, because you’ll be over 30″ high from 280 yards to 370 yards, and over 4″ high between 30 yards and 590 yards.  Not very useful.

What turns out to be the best zero in this instance, given the above criteria, is a 260 yard zero.  This puts me 3.95″ high at 140-150 yards and 4″ low at about 307 yards.  We can call 300 yards our maximum PBR for this load and target size.  If we change the criteria by dropping our standards and calling the target size 10″, then a zero at 285 yards will get us out to 337 yards.

To illustrate what a “flatter shooting” cartridge will do for us, consider the hypothetical 6.5-06 load I referenced above.  For an 8″ target, a zero setting of 290 yards will get us out to 343 yards.  If we increase our target size to 10″, a zero setting of 319 yards (yes I understand it’s a little ridiculous) will get you out to 377 yards with no adjustment or holdover.  A faster cartridge might extend the PBR a bit but there are other tradeoffs.

So if you are hunting big game and don’t want to fool with your sights or hold over, you will have to accept a practical limitation of shooting from less than 300-350 yards and embrace the PBR concept.  If you want to shoot farther than that, become a master of trajectory.  Know exactly what you’re bullet drop is, and your next limitation becomes your cartridge’s terminal ballistics.

Note that everything I’ve written concerning PBR has assumed perfect accuracy and precision from you and your system.  At 300 yards, the 8″ target is about 2.6 MOA.  At 200 it’s 3.8, and at 100  it’s 7.6.  At 300 in the cross leg sitting position, my group was about 2.4 MOA (7.6″), so using PBR I’d have a guaranteed hit, right?  No.  My group size of 7.6″ has to be added in with the distance my trajectory is already off from my line of sight.

If I was at the bottom of my “wobble area” in sitting, my shot at 300 yards would theoretically land 3.8″ (that’s half of my known group size) low.  Look at the trajectory diagram again.  At 300 yards, I’m already at the bottom of the trajectory, 4″ below my crosshairs.  Add in the wobble factor, and that makes me 3.8″ below the vital zone.  Between 85 yards and 245 yards, given my known group size, I could be just over 2″ high.  Think of it as “tolerance stacking” applied to field shooting.

What this tells me is that you need to know what you can hit, from what position, and from what distance with your zero before you go making assumptions.  The more attractive option, in my opinion, would be to know your holdovers well enough to do them under stress.

If you want to learn more about external ballistics, which is what this article was trying to be about, I highly recommend you buy and read Bryan’s book, Applied Ballistics for Long Range Shooting.  In fact, I recommend you buy my copy of the 1st edition at full price, so I can buy the 2nd edition.  I’ll throw in a dummy round as a bonus!!!

 

3 thoughts on “Trajectory

  1. Pingback: The Stair Supported Position | Art of the Rifle

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