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Old 10-13-2007, 10:37 AM
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Exterior Ballistics -- A Primer



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Exterior Ballistics -- A Primer

For the sake of discussion, let's start with a rifle locked into a bench rest and sitting on a sturdy table. The rifle is positioned in such a way that the barrel is perfectly horizontal. Let's also assume the barrel is completely immobilized, allowing us to ignore the effects of recoil. And let's also assume the sights (be they iron or the crosshairs of a scope) are set and adjusted in such a way that the line of sight is 1.5" above the centerline of the bore. For the moment, let's also assume the sights are aligned so as to be perfectly horizontal (and therefore perfectly parallel to the centerline of the bore). Finally, let's put a target out at a range of 100 yards (300 feet).

So we have something like this:




Now let's enter a strange world where the air is free from the effects of friction -- a bullet that takes off at a certain speed will keep moving at the same speed until it hits something, and we can safely ignore the wind (we could just call it a vacuum, but where would be the fun in that?). This same strange world is also somehow immune to the effects of gravity -- our bullet will move in a perfectly straight line until it hits something. Let's give that bullet an initial speed (muzzle velocity) of 1,000 feet per second. Since it is exactly 300 feet from the end of the barrel to the target, we know the bullet will hit the target exactly 0.30 seconds after it is fired. If we were to sketch out the trajectory of our bullet over time it would look something like this. Pretty boring, eh?





Enter Gravity

So far we have been looking at a bullet that has all its motion in only one direction, that being the horizontal plane. Adding gravity to the mix introduces motion in a second direction, that being the vertical plane. If gravity were simply a fixed velocity it would be no big deal. If, for example, our bullet is moving down at a steady speed of 32 feet per second at the same time it was moving horizontally at 1,000 feet per second, it will fall (drop) 115 inches by the time it reaches our target at the 100 yard range line.

But, as you know, the effect of gravity is not a steady speed and our bullet does not leave the barrel already moving downward at 32 feet per second. Instead, our bullet leaves the barrel with no speed in the downward direction but immediately begins accelerating downward at the rate of 32 feet per second per second. Given all that, the math says that by the time our bullet reaches the target at the 100 yard range line (0.3 seconds after it was fired) it is moving downwards at a speed of about 9.6 feet per second and has dropped only about 17.5 inches. So, does that leave us with a trajectory that looks like this?



No; it does not. Why not? Because of that pesky acceleration thing. The short version is that near the barrel of the rifle the bullet will fall only a little bit in a given period of time; but as the distance from the barrel increases (and the bullet has time to accelerate downward) it will fall greater and greater distances in the same given unit of time. We can use our example of a bullet with a muzzle velocity of 1,000 feet per second fired at a target 100 yards away to illustrate: In the first 30 feet of horizontal travel (which takes 0.03 seconds) the bullet will fall 0.014 feet and be moving at a downward speed of 0.96 feet per second. But in the next 30 feet of travel (which also takes 0.03 seconds) the bullet will fall 0.043 feet and accelerate to a speed of 1.93 feet per second. And in the next 30 feet of travel it will fall another 0.072 feet and accelerate to a downward speed of 2.89 feet per second. We can do the math to figure out the acceleration over time and end up with a table that looks like this:


Code:
Range..Time.... Speed.... Drop
............... of fall........
30.... 0.030.... 0.965.... -0.2
60.... 0.060.... 1.929.... -0.7
90.... 0.090.... 2.894.... -1.6
120... 0.120.... 3.858.... -2.8
150... 0.150.... 4.823.... -4.3
180... 0.180.... 5.787.... -6.3
210... 0.210.... 6.752.... -8.5
240... 0.240.... 7.716.... -11.1
270... 0.270.... 8.681.... -14.1
300... 0.300.... 9.646.... -17.4
(In this and following tables, range is given in feet, time is given in seconds, velocities are given in feet per second, and drop is given in inches. Note that since in this case the bullet immediately drops below the horizontal, all the drop values in this table are shown as negative numbers. If, as we will do later, we incline the barrel so the bullet's trajectory takes it above the horizontal we get positive numbers in the drop column to indicate the bullet has risen above the horizontal [or line of sight, depending on what one uses as a reference line] and then negative numbers once it drops back below our reference line.)

If we plot the horizontal and vertical travel shown in the table above on a graph, we end up with a trajectory that looks like this:





Now we are getting closer to reality, with a nicely curved trajectory that illustrates the effect of gravity on our bullet's flight path. Still missing is the effect of drag (friction, or air / wind resistance) on the bullet's horizontal motion as well as its drop (vertical motion). For that we will need to look at the idea of ballistic coefficient, but before we do let's take a tangent to look at the difference in the trajectory of a bullet moving at a speed slightly faster than our current example.


A Flatter Trajectory


What if we leave the barrel of our fictional gun firmly anchored so that it is perfectly horizontal and change only the muzzle velocity of our bullet? If we increase the speed from 1,000 feet per second to 1,500 feet per second will the bullet have more drop at 100 yards than the bullet used in our earlier example? Or will it have less drop? Intuition should give you the right answer, but let's do the math to illustrate. Again we will use range increments of 30 feet. Since our bullet is moving faster, we will have to change our time increments from 0.03 seconds to 0.02 seconds as that is the length of time it will take the bullet to travel 30 feet in our world without friction. Doing the math gives us this table:


Code:
Range..Time.... Speed.... Drop
............... of fall.......
30.... 0.02.... 0.643.... -0.1
60.... 0.04.... 1.286.... -0.3
90.... 0.06.... 1.929.... -0.7
120... 0.08.... 2.572.... -1.2
150... 0.10.... 3.215.... -1.9
180... 0.12.... 3.858.... -2.8
210... 0.14.... 4.501.... -3.8
240... 0.16.... 5.144.... -4.9
270... 0.18.... 5.787.... -6.3
300... 0.20.... 6.430.... -7.7
And there we have it. The faster bullet will hit the target at a spot far above the slower bullet. The table has already shown us that, but let's put it on our graph so we can compare the two trajectories. The trajectory for our 1,000 fps bullet is shown in gray and the trajectory for our 1,500 fps bullet is shown in red. Pretty significant difference, eh? And it is due completely to the ever so slightly longer period of time that gravity has to accelerate the slower bullet in a downwards direction.





The Friction Demon

So far we've been dwelling in that mythical world loved by physics teachers, where air is a frictionless medium and our bullet reaches the target moving at the same speed it was when it left the muzzle of our rifle. But it is not that way in real life so let's look at the effects of friction (air resistance) on our trajectory. There are two effects we need to consider:
- the deceleration of the bullet's forward velocity
- the change of trajectory induced by moving air (wind)

In our previous examples, the bullet moved down range at a steady speed. In The Real World friction between the air molecules and the surface of the bullet will cause the bullet to decelerate. And, just as with the case of acceleration due to gravity, the effect of deceleration increases over time. In this case, the effect is to increase the curvature of the trajectory as gravity has more time to accelerate the bullet downward. Also, moving air (wind) will deflect our bullet's trajectory left or right and, in some cases, cause its trajectory to flatten out (a tail wind) or curve more deeply (a head wind).

All of these effects are the result of drag. One tool used in comparing the amount of drag experienced by bullets of various shapes, sizes, and weights is ballistic coefficient. Essentially, the BC is a measure of how well our bullet moves through the air and resists the effects of friction and wind as compared to the "standard round" that was used way back when they were first developing the science of ballistics. The BC can be derived through empirical observation (firing the same bullet umpteen hundred times under varying conditions and measuring the results), or it can be approximated mathematically using the bullet's weight, length, diameter, surface texture, shape (form) and velocity. Yep, that's right -- the bullet's velocity also affects its BC. The exact same bullet will have one BC at velocities above the speed of sound, and another at velocities below the speed of sound. Fortunately for us, the folks who came before us have worked out the BCs of most of the bullets we use so it is simply a matter of scrounging around until we find them. Then all we have to do is plug their BC into our math.

Before we jump into the math (with the help of a ballistics calculator) let's go back to that idea that bullets behave differently at velocities above the speed of sound than they do at velocities below the speed of sound. At speeds of about 1,000 fps or less, our bullets act as though they are simply pushing their way through the air, shouldering the air molecules aside as they pass. But at speeds of about 1,200 fps or more, they don't slip so easily between the air molecules and actually compress the air in front of them as they try to bully their way through. As a result, the faster bullet experiences more resistance -- more drag -- than the slower bullet, causing it to lose more of its velocity than the slower bullet does, and to be deflected further by the wind than the slower bullet. The "cross over" velocity for this effect is the speed of sound, where there is a rapid change in the bullet's coefficient of drag. Since our faster bullets are slowing down when they go through this barrier somewhere down range, we can look at it as a sudden drop from high drag to low drag effects on the bullet's flight.

First, let's look at how drag changes our trajectory. I'll be using a ballistics calculator that works out the drag for me. We'll go back to our bullet with a muzzle velocity of 1,000 fps for this example and once again look at its speed and drop at intervals along its path to our target at the 100 yard range line. Since we need it to do the math, let's say this bullet is a 40 grain round nose bullet with a BC of 0.169. When examining the table below and comparing it with our earlier table, note how drag decelerates the bullet, causing it to take longer to get to the target and giving gravity more time to accelerate the bullet downward, resulting in more drop at a given distance.

Code:
Range..Time.... Velocity... Drop
30.... 0.030.... 987.8.... -0.2
60.... 0.061.... 976.3.... -0.7
90.... 0.092.... 965.2.... -1.6
120... 0.123.... 954.5.... -2.8
150... 0.154.... 944.3.... -4.4
180... 0.186.... 934.5.... -6.5
210... 0.219.... 925.0.... -8.9
240... 0.251.... 915.8.... -11.7
270... 0.284.... 906.8.... -14.9
300... 0.317.... 898.2.... -18.6
Now let's plot this trajectory on our graph. We will overlay it on our earlier graph from the days of frictionless air. The original trajectory is shown in gray while our most recent trajectory is shown in red. As you can see, while it can't be ignored, the difference created by drag is pretty small.



Now let's take another look at our faster bullet. Again we are going to change only the muzzle velocity of the bullet; the barrel will remain locked into its perfectly horizontal position and we will use exactly the same 40 grain round nose bullet we used before. For the moment, let's cheat and give this bullet the same BC we gave to our slower bullet. Here is the table:

Code:
Range..Time.... Velocity... Drop
30.... 0.020.... 1463.8.... -0.1
60.... 0.041.... 1428.5.... -0.3
90.... 0.062.... 1394.4.... -0.7
120... 0.084.... 1361.3.... -1.3
150... 0.106.... 1329.4.... -2.1
180... 0.129.... 1298.8.... -3.0
210... 0.153.... 1269.4.... -4.2
240... 0.176.... 1241.3.... -5.6
270... 0.201.... 1214.6.... -7.2
300... 0.226.... 1189.3.... -9.0
If you look closely at the table above you can see that our faster bullet has a greater rate of deceleration than our slower bullet did. Yes, the bullet is faster than our earlier example, and it stays faster all the way to the target; but the rate of change (think percentage) in the bullet's velocity is greater. Do some quick math and you can see it -- the slower bullet still has almost 90% of its initial velocity when it hits the target, while the faster bullet only retains 80% of its initial velocity. As you recall from our previous discussion (above) this is because our faster bullet experiences more drag because of the way it bullies its way through the air.

Now, before we plot this trajectory on our graph, let's recall that bullets will have different BCs at different velocities. The 40 grain round nose bullet we are using has a BC of 0.169 at 1,000 fps, but at speeds above 1,050 fps it has a BC of only 0.145. So let's redo the table using the "proper" BC. You'll note that while the change in BC does make a difference (about 0.3" at 100 yards), it is much less important a factor than is the change in drag as a result of the higher velocity.

Code:
Range..Time.... Velocity... Drop
30.... 0.020.... 1457.9.... -0.1
60.... 0.041.... 1417.1.... -0.3
90.... 0.063.... 1377.8.... -0.7
120... 0.085.... 1340.1.... -1.3
150... 0.107.... 1304.0.... -2.1
180... 0.131.... 1269.6.... -3.1
210... 0.155.... 1237.0.... -4.3
240... 0.179.... 1206.2.... -5.7
270... 0.204.... 1177.5.... -7.4
300... 0.230.... 1150.6.... -9.3
Now let's do some plotting. The graph below shows three trajectories for our 1,500 fps bullet. In gray is our original trajectory from the days of frictionless air. In blue is the trajectory that cheats and uses the higher BC of the slower bullet. And in red is the trajectory we actually expect from this particular cartridge based on a BC adjusted for its higher velocity. Chances are that between the resolution of the image I made and the resolution of your screen, the red and blue arcs are so close together you can't tell them apart. Again, we see that it is the extra drag that comes with higher velocity rather than BC that has the major effect at the speeds and ranges we deal with using our .22 rimfires.

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Old 10-13-2007, 10:41 AM
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A Nod to Aeolus

Sorting out the effect of wind on our bullet is a little harder, mainly because bits of it are counter-intuitive. For example, you might think that the slower bullet (which takes longer to get to its target) would be blown further off course than the faster bullet. After all, one might reason, the wind is pushing on the slower bullet for a longer period of time, so surely it must be blown further off course. But such is not the case. In fact, the exact opposite is true: the faster bullet will be pushed further off course than the slower one. And the difference is not just a little bit either. I'm going to use a ballistics calculator rather than work out the math here, but here are the numbers:

Example 1: 40 grain round nose bullet with a muzzle velocity of 1,000 fps (BC of 0.169) shot at a target 100 yards away in a 5 mph crosswind: 1.5" of wind drift.

Example 2: 40 grain round nose bullet with a muzzle velocity of 1,500 fps (BC of 0.145) shot at a target 100 yards away in a 5 mph crosswind: 2.7" of wind drift.

Example 3 (where we cheat a little and give the faster bullet the slow bullet's BC just to show again that speed and not BC is the main factor): 40 grain round nose bullet with a muzzle velocity of 1,500 fps (BC of 0.169) shot at a target 100 yards away in a 5 mph crosswind: 2.3" of wind drift.

So what's going on? Two things. The first is the physics of bodies in motion. You could almost say it is one of those relativity things. While we concern ourselves with the bullet's path over the ground, from the bullet's point of view it is about the path through the air -- air that just happens to be moving sideways from our point of view. One of the things that happens in this interaction is that the nose of the bullet slews itself around to point into the wind. Remind yourself that the bullet is not a rocket (there is no thrust coming out of its base) and you'll realize the bullet is going down range with its nose at a slight angle to its flight path. The result is a change in the bullet's rotational axis, and the same spinning that stabilizes the bullet now also applies some torque to its motion in response to changes forced upon it by the wind. With the bullet's nose pointed however slightly upwind, the bullet's drag becomes a component that deflects the bullet's trajectory downwind, resulting in what we popularly call drift. If the wind is in the right direction the spinning forces of the bullet can result in the nose ending up with a slight downward tilt, causing the bullet to hit higher than expected as the drag actually works against gravity to slow the bullet's downward motion. This is what we see when our shots hit "high and left" when shooting in wind coming over our right shoulder.

The second thing at work is something we have already mentioned -- drag. Because of the geometry involved in the bullet's trajectory through a wind (see above), drag turns into a vector that deflects the bullet downwind. And because the faster bullet has more drag than the slower bullet, it is deflected further downwind than the slower bullet. Anyway, it is pretty freaky stuff so I'll leave it to Aeolus and the pros who make ballistics calculators to sort it out. A good place to start if you want to know more is right here:

http://www.exteriorballistics.com/eb...s/article2.pdf

By the way, it is the increased effect of the wind on fast moving bullets that has most "match grade" bullets designed to move at slower speeds.

Sighting In

Before we close this primer let's take a quick look at the problem of sighting in the rifle so that we can hit what we are aiming at. Our current setup has the line of sight from the scope (or iron sights) exactly 1.5" above the centerline of the bore. In all the examples I've used I've measured drop from the centerline of the bore. But when using a ballistics calculator, and when trying to get your rifle sighted in, the drop is measured from the line of sight. Among other things, that means the bullet is already 1.5" low the instant it comes out of the barrel and your trajectory compared to your line of sight looks something like this:



Getting the line of sight and the trajectory to intersect at a given range is what sighting in is all about. And, from looking at the graph above, you should be able to see that what it involves is inclining the barrel upwards slightly so that the trajectory starts with the bullet going upwards to cross the line of sight. And, from looking at the curve gravity puts into the trajectory, you should be able to see that to get sighted in at what we consider a normal range you have to incline the barrel at such an angle that the bullet is actually coming downwards when it intersects the line of sight at our desired range. Here, for example, is what our slow bullet's trajectory looks like when our rifle is sighted in at 100 yards:



From your point of view, of course, you aren't moving the rifle barrel around. Instead, you are adjusting the sights to angle downward a bit. But once you try to get back on target your line of sight goes right back to where it was and the barrel gets inclined upwards.

Now if you are clever you have noted a short cut you can take when sighting in your rifle. You see where the trajectory crosses the line of sight on its way up? That is your short cut. If you sight in your rifle so it is "dead on" at the range where the bullet crosses the line of sight on the way up you will find it is also sighted in for the range where the bullet crosses the line of sight on the way down. That might save you some walking, and it might be the only way you can sight in a rifle if you are restricted to an indoor range.


Point Blank Range

"Okay," I can hear you saying, "all this ballistics stuff is good for the paper punchers, but I'm a hunter and it don't mean squat to me!" Yes; but no. It is a given that paper punchers look to milk the most accuracy they can out of their rifles and studying the ballistics of their favorite cartridge is part of that. But a guy looking to make head shots on squirrels might want to know a little bit about it too. And the hunter, especially, is likely to be interested in how the ballistics influence his point blank range.

Put simply, the point blank range is that envelope of distance in which the bullet never gets more than so far above or so far below the line of sight. The "so far" is something you define yourself. Hunters usually set its size based on the "critical zone" of the game they are hunting. A deer hunter might pick a circle of 5" diameter to represent the heart & lung area of a deer, so would use 2.5" above and 2.5" below the line of sight as his critical zone for working out his point blank range. But since we are talking rimfire here, let's say you fancy yourself a squirrel headhunter, and you reckon that if the bullet is no more than one-half inch above or one-half inch below your point of aim you should put the bullet somewhere on the squirrel's head... the proverbial "minute of squirrel" measure. What we want to know, then, is:
- at what range to sight in our rifle so that the bullet never gets more than 1/2" above our line of sight;
- what the minimum and maximum ranges are where the bullet stays within the 1/2" above or 1/2" below envelope. If Mr. Squirrel shows up within that range envelope, we just put the crosshairs on the center of his little noggin' and squeeze the trigger.
- how much hold-over or hold-under to use at ranges outside the PBR envelope.

I'm going to cheat and go right to building a table based on sighting in at the perfect range (having already used my graphics calculator to figure out what it is). Again, just for an example, we will use our 40 grain round nose bullet with a muzzle velocity of 1,000 fps giving it a BC of 0.169.

Code:
Range..Time.... Velocity.. Drop
30.... 0.030.... 987.8.... -0.5
60.... 0.061.... 976.2.... +0.2
90.... 0.092.... 965.2.... +0.4
120... 0.123.... 954.5.... +0.3
150... 0.154.... 944.3.... -0.1
180... 0.186.... 934.5.... -1.6
210... 0.219.... 925.0.... -3.1
240... 0.251.... 915.8.... -4.7
270... 0.284.... 906.8.... -6.4
300... 0.317.... 898.1.... -8.1
When we plot the table on our graph, we get this:



So what does all that do for us? Well, it tells us that if we sight in properly our PBR is the envelope between 10 yards and 53 yards -- this particular bullet will never be more than 1/2" above or 1/2" below our line of sight anywhere in that range band. It tells us that at about 35 yards our bullet is as high above the line of sight as it is going to get (and that being only 1/2", eh?). It tells us that to get there we need to sight in our rifle for "dead on" at 47 yards. It also tells us that we can save a few steps by sighting in at 18 yards and then just checking with a few shots at 47 yards to make sure we are, in fact, sighted in at the longer range.

And there you have it.


The Usual Caveats & Disclaimers

First, you may have noticed this is a revision of an earlier posting I made on this topic. Many thanks to RFC-member Digital Dan for his help in the section on wind drift, and for the links he provided to a wealth of ballistics related information available on-line. You'll find these included on the list below.

In all the graphs showing trajectories I have exaggerated the vertical scale to more clearly illustrate the differences between one trajectory and another. No; it's not cheating since all of the trajectories are done on the same scale. And, just in case you're wondering, it is a technique commonly used in graphic ballistic calculators.

There is no way the results from your individual rifle will perfectly match the trajectories shown here. Nor will the results from your rifle match the trajectories produced by any ballistics calculator. Why not? For starters, not all of them accept weather data (barometric pressure, altitude, temp, humidity) as inputs yet all of these factors will influence the trajectory. Even if your calculator does accept them, do you really know what the weather data is for the exact moment you pull the trigger? And is your barrel of exactly the same bore diameter as the test barrel? The same length? Do you hold it in exactly the same way each time? No? Then the best we can do is use the calculator to get a general idea of how our rifle and ammo combo will perform. You actually have to pull the trigger to see what will really happen.

Questions?

Quote:
Where did you get the BCs you used in the calculations?
I pulled them from this site:

http://www.exteriorballistics.com/eb.../22rimfire.cfm

Other sources will give different values. It is a shame the bullet makers don't give BC values for their .22 rimfire bullets like they do their centerfire bullets.

Quote:
Tell me about this BC changing with velocity thing. Do I have to recalculate the whole mess whenever the bullet slows down from, say, 1,050 fps to 1,000 fps and then again when it slows to 900 fps as it goes down range?
No. Use the BC that matches the bullet's muzzle velocity. The ballistics calculator you use will handle the rest of it.

Quote:
Okay, you keep telling us the faster bullet has a flatter trajectory and will hit higher on the target than the slow bullet. Why is it, then, that when I try it at the range the faster bullet is hitting the target below the slower bullet?
We certainly can't argue with your empirical observation, but we also can't blame the bullet for what you are seeing. The faster bullet does have a flatter trajectory and will hit above the slower bullet if you keep all other things equal. But the faster bullet can hit below the slower bullet if the barrel is not in exactly the same position each time the bullet leaves the barrel. Unlike our examples above, where the barrel magically was made completely immovable, your rifle recoils which causes the barrel to lift upwards (since you probably used a rest that resisted movement in other directions). The slower bullet will take longer to get out of the barrel than the fast bullet, very likely leading to the barrel being inclined more upwards when the slower bullet leaves the barrel than it is when the faster bullet leaves the barrel. That is only one possible explanation. The oscillation (vibration) of the barrel may be another, as the two bullets may create different patterns of oscillation in the barrel as they travel down the bore at different speeds, putting the barrel in a different position when the bullet leaves the barrel. Remember, too, that this oscillation is not in a single plane.... the end of the barrel doesn't just wiggle up and down; it rotates around an axis. We see the effects of this in centerfire rifles, for example, when a fast bullet prints not only higher than a slower bullet but also to one side or another of the center. It is also why when floating a barrel you have to give the barrel some "wiggle room."

Quote:
Why are you saying that the faster bullet has a flatter trajectory? If you fire the two bullets from a barrel angled upwards, the faster bullet will go higher than the slower bullet. By definition, the fact that it goes higher than the slower bullet means it is not flatter! Here is the graph to show you what I mean, with the slow bullet in green and the fast bullet in black -- see how the faster bullet goes higher!
Maybe we are talking semantics here. Yes, if we point our rifle upwards at an angle the faster bullet goes higher above the ground than the slower bullet. And if we defined "flatter" as "closer to the ground" then that would indeed mean the slower bullet has a flatter trajectory. But the measure of whether or not a bullet's trajectory is "flat" is the rate of change in the bullet's trajectory curve. Or, in different terms, it is a measure of how close the bullet stays to a straight line extended from the bore of the rifle. Between two different trajectories, the one that stays closer to a line extended from the bore is the flatter trajectory. Or, to put it another way, it is a measure of how far along the trajectory curve the bullet can move and still be going (more or less) in a straight line. Take a look at these two trajectories; which of them has the longest section that a reasonable person might call a "more or less straight line"? Which of them stays closer to a line extended from the bore of the rifle (the straight black line on the graph)?



The trajectory with the longest section that we can call a "more or less straight line" is the flatter of the two trajectories. As we can see, it is also the trajectory that stays closer to the line extended from the bore of our rifle. And, regardless of the angle at which they are fired, the faster bullet will always have the trajectory that comes closest to our definition of a flat trajectory. If we change the definition, such as by using proximity to the ground as the test, then all bets are off.

Quote:
I thought the shape of the bullet created lift, just like an airplane wing, and that is what causes it to move upwards. And then, as it loses momentum, it starts to fall back down. Doesn't the shape of the bullet cause it to rise up as it spins and hits the air molecules, even if the barrel is kept horizonal?
Not exactly. While there is a tiny bit of lift created by the way the bullet's axis of rotation is always slightly above its flight path (an effect of tractability), the bullet itself is symmetrical in shape. On the other hand, airplane wings create lift because they are shaped differently on the top than they are on the bottom. The difference in the shape creates a low pressure above the wing and a high pressure below it. Together these forces act to push and lift the wing upwards. While the lift generated by the wing's shape is significant, the lift experienced by the bullet is very slight.

As we have seen above, gravity starts acting on the bullet as soon as it leaves the barrel of our rifle. And since the bullet can't create enough lift on its own to get up to our line of sight, and keep it high enough to hit the target, we have to angle the barrel upwards to give the bullet its head start in the fight against gravity.

Quote:
Where can I find a ballistics calculator?
You can buy software that does the job from several sources. The advantage there is portability -- you can take your laptop to the range and have at it. If you prefer to work online there are some available free of charge. My favorite graphic version is the one hosted by Norma. It doesn't take all the environmental variables into account, but if you are after a "quick & dirty" visualization of your trajectory it will do nicely. Here is the link:

https://www.norma.cc/en/Ammunition-A...stics-program/

You'll have to use the "define your own bullet option" to give it the stats for your .22 rimfire bullet.

If you want to crank in the environmental variables to see their effects, and you like seeing numbers instead of graphs, a pretty snazzy tabular version is available here:

http://www.jbmballistics.com/ballist...culators.shtml

For some ideas on what you can do with these things, do some reading here:

http://www.gunsmoke.com/guns/1022/22ballistics.html

Quote:
Where can I learn more?
There are tons of material available on-line. I've found some of the most helpful to be:

http://www.exteriorballistics.com/ebexplained/index.cfm

http://www.frfrogspad.com/extbal.htm

http://www.rifleshootermag.com/shoot...303/index.html

.
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Last edited by Sophia; 09-05-2018 at 05:56 PM. Reason: updating links.
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Old 10-13-2007, 10:46 AM
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A dame with brains.
Sophia, you really class up this joint.
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Old 10-13-2007, 12:07 PM
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That would have taken me about two weeks.
Nice work Sophia.
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Old 10-14-2007, 05:52 AM
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Curious. I am certain that I replied to this thread. However, my reply seams to have vanished. Does anyone else see it or is it just my computer?
Gobbler
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Old 10-14-2007, 06:49 AM
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Thanks a bunch! You just saved me a lot of walking back & forth to see where I need to sight in my SS ammo to get the longest PBR outta my squirrel rifle. I was sighted in at 25 which gave me DeadOn again at 40 but about -3/4 at 50 while the HV stuff was still DO at 50. I believe my scope is mounted a little lower though but I'm gonna try to zero at 18 and see how much it extends my effective range with SS while still holding DO!
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Old 10-14-2007, 10:35 AM
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Quote:
However, my reply seams to have vanished. Does anyone else see it or is it just my computer?
It's not you, Gobbler. This is a new thread with some additonal info in the section on wind effects.
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Old 10-14-2007, 06:50 PM
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That's good. For a minute I thought my time machine hiccuped again.
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Old 10-14-2007, 06:54 PM
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Another great informational write up Sophia, great stuff, thanks for the effort you expend for all of us.

Gerald
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Old 10-14-2007, 08:54 PM
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Where wus you

Child where wus you when some of us old farts was trying to figure things like max range, rate of twist, how far a 15 grain bullet will travel, etc.
Good work, too many shooters never take time to see just how much you can learn from doing the math even on a 22 rimfire. A little math can save a lot of guess work, just the squirrel hunters alone could buy your lunch for the POA advice. Keep up the good work young lady, and stay away from the old guys that are after you for your good looks and brains.
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Old 10-14-2007, 09:38 PM
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Not an issue of semantics....

Quote:
Why are you saying that the faster bullet has a flatter trajectory? If you fire the two bullets from a barrel angled upwards, the faster bullet will go higher than the slower bullet. By definition, the fact that it goes higher than the slower bullet means it is not flatter! Here is the graph to show you what I mean, with the slow bullet in red and the fast bullet in blue -- see how the faster bullet goes higher!
Quote:
Originally Posted by Sophia View Post
...Maybe we are talking semantics here...
Clearly, even in the illustration, the faster bullet does exhibit the flatter trajectory.

The error is that the two examples have different points of aim/impact (540 and 900 feet). "Flat shooting", to my mind, is more a term of comparison between two bullets dialed in to the same POI.

Thanks for the primer, Soph. Very interesting.

-S

Last edited by Sophia; 11-09-2017 at 06:55 AM.
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Old 10-15-2007, 12:01 AM
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The error is that the two examples have different points of aim/impact (540 and 900 feet).
Not so much an error -- but done to address the argument that was based on bullets fired at exactly the same barrel elevation angle. If the elevation is constant from one cartridge to the next, the "sighted in" range has to vary based on MV.

Quote:
"Flat shooting", to my mind, is more a term of comparison between two bullets dialed in to the same POI.
That would have the barrel at a different elevation angle for each cartridge, of course, and we would get something like this:



Where the faster bullet is shown in black and, again, has a flatter trajectory because it stays closer to our frame of reference (the line of sight) and has a "straighter curve" than does the slower bullet. This time, of course, the faster bullet is also closer to the ground but that really isn't relevant.
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Old 10-15-2007, 01:02 AM
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For everything I see Sophia I will agree with you. But now I'm going to mess with you in the REAL world.

To best illustrate this effect we are going to go to a common extreme....the .44 magnum in a hand gun.

You are going to shoot two loads, both with 240 grain bullet.

One load will have a MV of 900 fps

One load will have a MV of 1400 fps

Range is going to be 25 yards.

Which bullet will hit the lowest on the target?

Contrary to what you have posted the SLOW bullet will hit higher on the target and the fast load will hit lower WHY? WHY in the real world does this happen?

Two related answers Recoil and Barrel Residence Time. Recoil stats the instant the powder is ignited and bullet starts to move down the barrel. The faster load reaches the muzzle before the barrel has recoiled quite as far so it leaves at a shallow angle. The slower bullet is in the barrel for longer time. The barrel has traveled farther in it's recoil arc before the bullet leaves the barrel and the barrel is at a steeper incline when the slower bullet leaves the barrel. It ALWAYS strikes the target, at relatively close range, higher on the target.

Does this only happen in large bore handguns?

NO

Load a 180 grain bullet in a .308 Win at two different velocity and you have the same thing out to a certain range.

This is the real life part the ballistic charts leave out. Recoil and barrel residence time.
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Old 10-15-2007, 05:21 AM
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Quote:
Originally Posted by Vincent View Post
For everything I see Shophia I will agree with you.
"Shophia"?
Vincent, are you hittin' the sauce again?
You're slurring your words there, pal.
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Old 10-15-2007, 08:13 AM
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BBQ Sauce I don't drink much. But if you look at the time the post was made..............
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