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Basic Bullet Trajectory Explained

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DairyFarmer View Drop Down
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    Posted: September 10 2014 at 1:23pm
A good place to start is to read the following Wiki article

http://en.wikipedia.org/wiki/Rifleman's_rule

In its most basic form, you need to know the following:

Line of sight (LOS)
Horizontal distance to target
Bullet path



LOS is the path to the target that is seen, through the sight, to the target. This must not be confused with the bullet path. The LOS is not affected by gravity, slope and other environmental factors. (Quantum scientist please excuse me.)

Bullet path is the path the bullet takes over a distance. This is not parallel to the LOS as the bullet is affected by gravity, horizontal distance, curve of the earth and earth rotation. Yes wind, humidity, temperature, air density, etc. do affect it too. But these, along with curve of the earth and earth rotation, are minor factors for now. So lest just say the path is a curve. As soon as the bullet leaves the barrel is starts dropping below the end of the barrel. Bullet shape and weight (BC and SD), rate of twist and speed of the bullet all factor into the rate the bullet will drop. Slower, heavier bullets drop quicker than light, fast bullets.

For the simplicity of this post, I will leave out environmental factors for now, i.e. wind, air density etc.

And then there is the horizontal distance to the target. This should not be confused with the distance over ground to a target. The steeper the slop, the less the horizontal distance is in relation to the distance over ground to a target. Say you were shooting at a target down/up a 45 degree slope. If you paced it out and it was 280m, you would actually only be shooting at a target 200m away (Pythagoras theorem). That is because gravity is only affecting the horizontal distance.

So how does this help us? In order to sight in a firearm we need to know where the bullet will be at a set of distances. Knowing what a bullet does in flight, its trajectory, will help us know where to expect the bullet to be.
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DairyFarmer View Drop Down
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Post Options Post Options   Thanks (0) Thanks(0)   Quote DairyFarmer Quote  Post ReplyReply Direct Link To This Post Posted: September 10 2014 at 5:16pm
Why does a bullet appear to rise and then fall when displayed on a ballistic chart?

If we take a line from the end of the barrel to the target, the bullet is always travelling downwards. Gravity is acting on the bullet.

Sights are mounted above the barrel. Because the rear sight is higher than the front sight, the LOS is angled downwards as apposed to the line from barrel to target.

Because the bullet is travelling in an arc, at very close range it will strike lower than the LOS (0 - 20m as an example). Gradually the two will come together and strike at the same point (example 25m). This is your first point blank range. As we get further out the bullet path is now higher than LOS (example 30-90m). Now the two lines will meet again (example 100m). This is you second point blank range.

As your second point blank range moves further out, so too does your first point blank range. The arc above the LOS may now be more exaggerated.

So lets say you want to sight in at 100m. The figures below are not actual figures but just an example.
0-10m the bullet may strike 0.5 to 1" low
25m 0"
60m 1 to 1.5" high
100 0"
200m 2" low
300m 5" low

As you can see the further away the quicker the bullet drops.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Zed Quote  Post ReplyReply Direct Link To This Post Posted: September 10 2014 at 8:34pm
Thank you Dairy Farmer for taking the time to post this subject.
While I already know some of the basics; learnt from my Father when I was a kid. But it's refreshing to see it explained clearly and to go a bit deeper into the subject. For example shooting up or down hill, it's comon sense really but not something you would obviously think of; but important  to know if hunting.

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Post Options Post Options   Thanks (0) Thanks(0)   Quote DairyFarmer Quote  Post ReplyReply Direct Link To This Post Posted: September 11 2014 at 8:53pm
Ballistic coefficient (BC) is the bullet's ability to overcome air resistance. In other words drag. The higher the BC the better!

A high BC bullet gets to the target faster and with more energy than a bullet with a low BC. Less time means the bullet is affected less by gravity and environmental factors. So the trajectory is flatter.

Sectional density is a how well a bullet's shape overcomes resistance. A spitzer bullet needs less power behind it than a round nose bullet. SD is used to calculate BC.

So what does this mean as far as bullet trajectory.?
If you can reduce the drag on a bullet it will shoot flatter and will travel further. Hence the arc is less pronounced.

If you reduce the power needed for a bullet to penetrate then you can use less power to get the same penetration, or you can use the same power to get more penetration. And by penetration we also mean drag because the bullet is penetrating the air. So we can get the bullet to travel faster and thus less time to the target. Less time for gravity to affect the bullet.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Zed Quote  Post ReplyReply Direct Link To This Post Posted: September 12 2014 at 1:54am
So The boat tail Sierra Match kings would have a higher BC than a flat base standard MkVII bullet even if the weight is the same because it has less drag.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote DairyFarmer Quote  Post ReplyReply Direct Link To This Post Posted: September 12 2014 at 2:40am
That is correct.

Sierra says that a 174gr HPBT Match has a BC or .499 @ 2200 fps and above, .493 between 2200 and 1800 fps, and .480 @ 1800 fps and below.

A MkVII 174gr pointed bullet with a flat-base has a BC of .467 @ 2440 fps.

So the Sierra travelling 240 fps slower has a BC of .032 more.

Just as a comparison a 7x57 (.284) with a Sierra 175gr HPBT has a BC of .608 @ 2100 fps and above, .582 between 2100 and 1530 fps, .532 between 1530 and 1300 fps, and .500 @ 1300 fps and below
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Post Options Post Options   Thanks (0) Thanks(0)   Quote Gun Nut 4 Quote  Post ReplyReply Direct Link To This Post Posted: December 21 2015 at 4:16am
Barrel vibration is also factor, slow bullets exit the barrel at higher points in its oscillation, while faster bullets exit the barrel at lower points in its oscillation. I use this factor to coordinate my big game and small game loads such that they both cross the LOS at the same initial point. I use a 185 or 205 grain R/N gas check cast bullet for big game, and 83 grain W/C or 93 grain R/N  plain base cast bullet for small game, they are great for close in shots at rabbits, squirrels and grouse.  
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Post Options Post Options   Thanks (0) Thanks(0)   Quote DairyFarmer Quote  Post ReplyReply Direct Link To This Post Posted: December 21 2015 at 4:51am
Barrel oscillation is not that important in trajectory as it affects accuracy more than it affects bullet path.
 
Sight height, bullet weight, bullet shape, velocity and distance to target are the key elements to trajectory. Trajectory is very much based on theory or potential. Accuracy is based on all the rest of the factors that the shooter controls; it load consistency and matching to rifle, windage, oscillation (which includes bedding and floating to obtain the correct harmonics), time between shots (temp affects the rifle more than any other factor), trigger pull, .................
 
I have just reread my OP and realised I left out something important regarding steep slopes.
 
You must also take into account where the vitals are situated. For example shooting up a steep slope, if you hit the animal in the right side you may shoot over the heart and the bullet path will be high on the left side. Although this is not as much a problem as raking shots on the flat. Most hunters would rather go for a high neck shot so as not to destroy half the meat. Raking shots should be left for cases where a wounded animal is fleeing or charging.
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Post Options Post Options   Thanks (0) Thanks(0)   Quote pogson Quote  Post ReplyReply Direct Link To This Post Posted: July 12 2018 at 6:09am
As much as I respect dairy farmers, I must correct a few things in DairyFarmer's post:
"If we take a line from the end of the barrel to the target, the bullet is always travelling downwards. Gravity is acting on the bullet."
It's possible to launch a bullet that way but it's unlikely except when shooting downhill. In the usual case of firing at a target at a long range, the bullet must be launched upwards somewhat to counter gravity. A baseball thrown from the outfield to home plate must be launched upwardly somewhat for the same reason. Gravity would cause the ball to hit the ground short of the target otherwise.

"Travelling" is a vague term here. One should refer to the height above ground or line of sight or speed and direction. For long range shooting for hunting, a bullet is usually launched about 8 minutes of angle above the horizontal. For 1000 yard shooting that angle has to increase to about 30 minutes of angle or the bullet will fall short. At the muzzle, gravity is acting downwards but it acts to slowly reduce the upward component of velocity, not the height of the bullet.

"Because the bullet is travelling in an arc, at very close range it will strike lower than the LOS (0 - 20m as an example). Gradually the two will come together and strike at the same point (example 25m). This is your first point blank range. As we get further out the bullet path is now higher than LOS (example 30-90m). Now the two lines will meet again (example 100m). This is you second point blank range."

Clearly, DairyFarmer knows what's happening. It's just a lack of clarity in expression. I also object to this use of the term "point blank range". I call it a "zero", when the sights are right on the expected point of impact. "Point blank" actually means close enough that the sights don't have to be used or that the point of impact will be close to the point of aim for a range of distances to the target. This is a very important concept for hunters as the vital zone of a deer may be nearly 12 inches (spine, lungs, heart). To be conservative and allowing for less accuracy in the field, let's say the vital zone may be 8 inches. Some use a point blank range that lets the trajectory fall within 4 inches on the high side and 4 inches on the low side and an aiming point in the centre of the vital zone. That's wasteful of the capability of a rifle however as a bullet falls much more rapidly after the second zero than before. That's because the bullet is slowing down due to air-resistance. I use a zero with 8 inches above the line of sight and I aim at the 6 o'clock position on the vital zone.

Consider a .303 150 grain pointed bullet with a ballistic coefficient of 0.3 and 2700 ft/s muzzle velocity. That has an 8 inch height of trajectory above the line of sight to 315 yards with remaining energy of 1115 ft-lb and 1840 ft/s impact velocity (bullet should expand and still penetrate). Using the +/- 4 inch rule, the maximum point blank range is only 285 yards with a second zero of 244 yards. The bullet is 4 inches high at 135 yards and 4 inches low at 285 yards relative to the line of sight. So the +/- 4 inch rule gives up 30 yards for no good reason unless the bottom of the vital zone is obscured by long grass/brush. My deer have long enough legs to eliminate that problem. I've only needed this extra range a few times in my life but it brought home a few good deer. It works for my son too, although he's into these new-fangled high-velocity rifles. The concept is the same whether the maximum point blank range is 300 yards or 400 yards. I like having available the full potential of 303 British even if it will rarely be used. Further, military snipers can do the job at much longer ranges by holding even lower, say down to the belly button for 600 yards... For hunters the old 303 will do the job out to as far as most can see a deer.

So, thanks for your contribution, DairyFarmer. It's very useful but I just wanted to clarify a few points.
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