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:
![Image]()
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?
![Image]()
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?
![Image]()
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:
(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:
![Image]()
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:
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.
![Image]()
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.
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.
![Image]()
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:
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.
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.
![Image]()
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.... -17.4If 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
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 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.... -9.0Now, 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
man: 
