Range Testing Theory for Spud Guns (Or any Projectile)

Going through my files, I came across this discussion that I wrote about the science behind spud guns (For actual guns that I’ve built, check out this mini-bolt-action model or this double-barrel pneumatic air cannon). If you’re interested in calculating the range of a projectile thrower of any kind, this discussion may be of some use to you.  I don’t recommend building a spud gun, and this whole discussing was written while I was still in college, so keep that in mind (also, where you see: “X^2” or “V0^2” etc that means X or whatever variable squared):

Spud Gun Testing and Ballistics

One thing that applies to both pneumatic and combustion spud guns is that the range of your cannon is related to the muzzle velocity or velocity at the end of the barrel. When a potato –or your choice of projectiles- is shot from your cannon, it has a certain velocity. This gives the potato some kinetic energy which hurls it away from the earth. Unfortunately, there is another force acting on your potato known as gravity. This force pulls your potato to the earth with an acceleration of 32.2 Feet per second per second or 32.2 Ft/s^2. To put it simply, if your potato was shot straight upwards at a rate of 100 FPS (feet per second), it would decelerate at a rate of 32.2 Ft/s^2. So after 0 seconds it would be going upwards at a rate of 100 FPS, after 1 second it would be traveling upwards at a rate or 77.8 FPS, and after 2 seconds it would be traveling upwards at a rate of 45.6 FPS. This trend continues and finally makes the potato fall to the ground in a similar manner. This all can be put into an equation like so:

(3.) Projectile Velocity = V0 –G*t

Where: V0 is the muzzle velocity out of your gun
G is the gravitational acceleration constant (32.2 Ft/s^2)
And t is the time from when the potato was first launched.

One useful thing about this equation is that when you set projectile velocity equal to zero, the projectile is at is apex, or the maximum height that it will achieve in its flight. A useful fact is that a bullet or other projectile fired directly upward will come down with the exact same velocity when you neglect air resistance. So, it will take almost exactly as long for your potato to reach the ground as it did to reach the top of its flight path.
This brings up one (not very good) way to figure out the muzzle velocity of your potato gun. In theory, if you were to shoot your gun straight up in the air and time how long it took for the potato to come down you could use equation 3 to figure out what the muzzle velocity was. To do this, one would set the projectile velocity equal to zero –the potato velocity at its apex-, and take t to be the time at which the potato hit its apex -how long it took the potato to come down divided by two. These two values would be put into equation 3. When this is all solved it would give the result:

(4.) V0 = G*t

Also included is a handy chart based on this equation. Just plug in the time it took for your potato to drop, and it gives you the muzzle velocity. Man, we do all the work for you. Interestingly enough, this chart could also be used to figure out how fast a ball comes off a kicker’s foot in a football game, or how fast you can throw a baseball up in the air. Incidentally, this only corresponds to whatever velocity is in the vertical direction, the rest will be explained shortly. Also, remember that this does not take into account air resistance, so your muzzle velocity will actually be a little faster than in the table. If you want a quick and dirty way to find your max range and could care less about the theory behind it, just get the muzzle velocity off this chart and then skip on to page 15 and plug it into Table 3.

This method isn’t, however, the best way to find the muzzle velocity of your potato cannon for two reasons. First of all if you shoot a potato into the air it is going to come down somewhere. The chances of it landing on you are pretty slim, but they are much greater if you decide to shoot it straight up in the air. Consequently, I don’t recommend this method. The second reason is that it is pretty hard to get an accurate measurement using this method. Air resistance takes it’s toll on accuracy, especially at higher velocities, and it would be hard to time exactly when one shoots the cannon and exactly when the potato hits the ground. Not to mention that it’d be hard to find the actual potato while dropping.
So there has to be some better way of measuring how fast your spud gun shoots. Another way, based on the same principles described above would measure how far a potato dropped in a certain distance. From this, you could infer how fast your potato was going, however there is the issue of measuring accurately how far you are shooting, and how far the potato dropped in that distance. This provides some practical difficulties, because it’s pretty hard to know if where you are shooting is completely flat, but would probably be more accurate if set up correctly than firing your gun straight into the air.
If you would like to get some estimate of your muzzle velocity in this way, set up your potato launcher on a flat surface as shown in Figure 2. Remember, the barrel must be as level as possible, and the ground should be flat. Otherwise, the measurements taken will be pretty much worthless. A laser level would be a good tool to use here, just to make sure everything is up to par.

From this setup, you can figure out how far the potato drops. You know how far the potato traveled, so the thing you need to find is how long the potato hung in the air so that gravity could pull it down. Another useful equation –we’ll call it the position equation- is needed:

(5.) Position = V0t + 1/2at^2

Solving this equation for t (and since a –acceleration- is zero) we get:

(6.) t = Distance/V0

Ok, that’s not too useful when t is not known. If you apply equation 5 in the vertical direction (V0 is equal to zero now because initially the potato isn’t moving in the vertical direction, and a is equal to g or 32.2 ft/s^2), and solve it for t, you can find out this time value directly from the two known distances:

(5.) Position = V0t + 1/2at^2

In this case: (7.) Position = 1/2gt^2

Therefore: (2*measured drop)/g = t^2

And: (8.) t = sqrt(2*measured drop/g)

Once you know the time, the last part is easy. Using equation 6:

(9.) V0 = distance to target/t

So there you have it, using equations 8 and 9 you can find the muzzle velocity of your gun. Just get time from equation 8, and then plug it into 9. One thing to note is that your units must be consistent. In other words, you can’t put inches and feet in these equations together and expect them to come out correctly. If you decide to use inches, you must use 386.4 inches/second^2 instead of the usual 32.2 feet/second2. Also, height and distance must be in the same units.
One thing to note about this method is that since the distance the potato goes is typically much less than that of a potato fired up into the air is that wind resistance or drag will have much less of an effect on it, so in that respect your results should be more accurate. Also, as in table 1a, you can directly read numbers off of this chart and then put them into table 3 to find your gun’s maximum range.

There are at least two other ways to measure muzzle velocity – I’m sure there’s more. First of all you could use a radar gun, pretty good solution since that’s what they are made to do – measure velocity. But where are you going to get a radar gun? I mean I guess you could buy one somewhere, but that’s probably pretty expensive. What about those annoying signs that tell you that you are speeding? Why not take a shot at one of those? It’s obliged to tell you the speed, so why not kill two birds with one stone? No, no, I’m not recommending that – it’ll probably get you thrown in jail, so if you want a really accurate measurement, just pony up the cash and buy a radar gun.
Another setup that could be used would act something like this: have two sensors, most likely infrared, that send a signal to a timer telling it whether or not there is something in it’s path (a potato for instance). The distance between these two sensors is known, and the timer starts when the first sensor picks up a potato (figure 3 illustrates this). This timer is stopped when the potato is sensed by the second sensor. So the time it takes to cross that distance will be found. Distance divided by time equals velocity, so this is a pretty easy calculation.
I’m sure it’s possible to build a setup like this one, but it would be fairly expensive. Sensors that can to what would be needed would cost quite a bit, and hooking up a timer wouldn’t be a piece of cake either. If you really want to make this setup, a good place to start would be to look at industrial controller suppliers such as Automationdirect.com. These sensors are known as “light curtains”, and you should be able to find some sort of timer to set up with it.
I believe that some paint ball shops have a setup for measuring the muzzle velocity of their guns as well. This might also be a good place to get your gun tested if they have the right equipment.

So what? Who cares if you know the muzzle velocity of your spud gun?  Well there is one other advantage other than bragging rights, the range of your cannon can be calculated from this muzzle velocity. Also, you can adjust your angle of fire so that you can hit something closer than your maximum range if you understand how this works.
If you have ever used a spud gun, or anything like that, or even if you’ve played most any sport, you’ve undoubtedly noticed how thrown or shot things fly through the air. In order to make a baseball go farther you have to lob it up in the air; in order to shoot a potato cannon a great distance you also must throw it up into the air. This is to compensate for the bullet – drop experienced by the potato; if the potato isn’t thrown into the air it will hit the ground prematurely. Look at Figure 2 for an example of this.
Mathematically, this is a result of equation 7. In this case, though, equation 7 has to be applied in both a horizontal and vertical direction –let’s say x and y direction for simplicity. When you shoot a gun out of a cannon, the velocity isn’t straight up in the air, as in equations 3 and 4, and it’s not horizontal as in equation 7 either. It’s some mix of the two.
So, how does one differentiate between the horizontal and vertical components of velocity? You have to use the sin and cosine relations –relax they are in your calculator.
Just use equations 10 and 11 to find each component, and you’ll be on your way.

(10.) Vx = V0*cos(angle)

(11.) Vy = V0*cos(angle)

Where: Vx is the muzzle velocity in the horizontal direction
Vy is the muzzle velocity in the vertical direction
Angle is the angle the gun is pointed (above the ground)
V0 is the muzzle velocity of the gun

Table 2 relates the relative muzzle velocities in the x and y directions. If you don’t have a calculator handy, a rough estimate can be obtained for each velocity component by multiplying your muzzle velocity by the relative velocity read off the chart. As you can see, the velocity in the x direction is zero when the gun is pointed at 90 degrees (straight up). The velocity in the y direction is zero when the gun is pointed level (at zero degrees).

So now you know how to separate out the two components of velocity, the next step is to turn them into something useful. The thing to think about here is that gravity only works in the vertical direction. You know, what comes up must come down, not everything that goes side to side must come back, come on. So now that you’ve got your mind right, all you gots to do to figure out how long your potato will be in the air is to apply good old equation 5, the position equation in the y direction only. Check it out:

(5.) Position = V0t + 1/2at^2

Sooo… (12.) Positiony = Vyt + 1/2ayt^2

Just remember, in this case ay is equal to the acceleration of gravity or 32.2 Ft/s^2. When the position is equal to zero, the potato has hit the ground (naturally). Since you know everthing else about this equation, you can easily solve for the time it takes to hit the ground. This is given in equation 13.

(13.) t = sqrt(Vyt/16.1)

So, since we know how long the potato will be in the air, equation 5 can be applied in the horizontal direction. Since the acceleration is zero in this direction, the position equation in this case works out to be:

(14.) Postionx = Vxt

This is the distance that your potato travels once it has left your gun. Now that wasn’t too hard, was it? Combine this with some sort of rangefinder, and with a little practice, you’ll be hitting whatever you want. Below is a trajectory chart for a gun fired at 200 FPS. Notice that the greatest range is obtained when fired at a 45 degree angle. This is a pretty useful rule, so keep that in mind.

There is also a very simple equation that can be derived from the formulas above.  I’ll call this the simple range equation.

(15.) Simple Range Equation:

Range = V0^2*sin(2*angle)/g

Where: g is gravitational acceleration (32.2 Ft/s2)

Using this equation, or equations 13 and 14 if you prefer, this maximum range chart was generated. Used in conjunction with Table 1a, the maximum range for your gun can be estimated without even using a calculator. This is all figured at 45 degrees, which, as mentioned earlier, is the angle you want to point your gun for maximum range. Just use a stopwatch to time a potato’s fall and you should be close enough for government work as they say.

What about drag you ask?  How much does it affect the projectile’s range and speed? Unfortunately, the answer to this is pretty complicated.  If you’re using something pretty dense, like a cannonball, maybe it hardly will.  However if you are using something less dense like a potato, the effects of friction will be greater.