# How high does my shell go

### From PyroGuide

**A shell altitude calculator**

One of the more common problems for amateur pyros, particularly those who make their own black powder, is calibrating shell lifts. We can measure the time they take to reach apogee (the highest point) so we know how long to cut the fuse, but we can’t easily measure how high they go. There are tables, certainly, and rule-of-thumb formulae, but they mainly apply to large, heavy shells for which pyro professionals have done a great deal of experimental work – not all of which they are willing to share. There’s nothing much to help out with small shells like festival balls.

One way to determine shell height is by triangulation. Basically this involves making a kind of sextant – we can use something as basic as a pivoted cardboard tube on a cardboard quadrant – and watching the shell’s flight through the tube from a measured distance away, say 200 feet. We pivot the tube upward to follow as the shell rises, and measure the angle from the horizontal at maximum elevation. From this we can use high-school geometry to calculate the maximum height. The disadvantage is this is takes great deal of work by a team, although it is probably the most accurate method.

The way most amateurs do it is to shoot a dummy shell and, using a stopwatch, time how long it takes to fall back to earth from its maximum height (apogee), which can then be calculated. We can use golf balls or baseballs for this, depending on the caliber, or for greater accuracy make some real shells filled with sand to the correct weight. One person can do this alone, though it’s helpful to have a second finger on the stopwatch. Most people know the simple formula, based on Newton’s 3rd Law of motion –

Height in feet = 16 t^2

Where t is the time it takes the shell to fall from apogee, in seconds. Unfortunately, while this is fairly accurate for large heavy shells, it becomes increasingly inaccurate as shells become smaller and lighter. The reason is the effect of aerodynamic drag, which affects light objects much more than heavy. If you doubt this, try throwing a golf ball and a ping pong ball up in the air and see which one you can throw highest.

There is no simple formula relating fall time to height that takes air resistance into account. Such a formula can be worked out, but it doesn’t result in a neat and simple relation like the equation above - it’s a 2nd order differential equation that’s very challenging to solve analytically. It’s much easier to solve such equations numerically. I’ve made up a little spreadsheet calculator that uses numerical methods to generate a table for any given size and weight of shell. Weigh and measure your shell, plug the numbers in, and Excel will spit out a table of heights and times. Now you can shoot a shell, time its fall with your stopwatch, then look up the time in the table and read off the height. Don’t have Microsoft Office? Then download and install the free Open Office (www.openoffice.org) from Sun Microsystems, and open the spreadsheet in Calc. That’s what I used.

These are the parameters you plug in. Measure and weigh your shell – just the shell, not the lift – and enter the diameter in centimeters and weight in grams in the boxes outlined in green above. I’m sorry about using metric units, but you'll get used to it. One inch is about 2.5 centimeters, and one ounce is about 28 grams. These are rough conversions, but accurate enough for this job.

Initial V and Initial H should both be set to 0. We are starting from the apogee, where the shell is neither rising not falling, so its velocity V is zero and the distance it fell so far is also zero.

Choose a Time Increment, in seconds. You can set it to 0.1 for tenths, or 1 for whole seconds, or anything else you like. 0.2, one fifth of a second, is a good compromise, but play with it if you like. The spreadsheet produces one line of output for every time increment.

The Drag Coefficient is a tricky number. It’s been calculated fairly accurately for most shapes, but for a ball it’s highly variable, anywhere from 0.05 to 0.5. The problem is it varies with speed, but it’s used to calculate the speed, so it’s a circular relationship. When the ball is moving slowly it’s low, and when it’s moving faster it’s high. As the shell leaves the mortar it’s very high, possibly greater than 0.5. Fortunately a great deal of work has been done on golf balls, and for the range of speeds we’re concerned with, a golf ball has a drag coefficient of 0.3. This may be a little low, as golf balls are carefully engineered for minimum drag. Perhaps 0.35 would be more typical of a papered shell. You can play with this value and watch how it affects the results.

If you haven’t used a spreadsheet program before, you’ll notice that as you enter your numbers, everything else on the page changes as the program recalculates based on your new values. **Be careful not to change the numbers in any cells except the ones outlined**. When your values are placed, the table is calculated ready to read or print.

I ran a sample calculation with the time increment changed to 0.5, half-seconds, to get all the results on one page so that I can explain them.

The first column is the elapsed time, in seconds. The second is the speed the shell is falling at that time, in meters per second. The third column shows the distance it has fallen in meters, and the 4th column is the distance converted to feet. The fifth column shows the acceleration, the rate of change of speed, in meters per second squared.

You will notice that after about 7 seconds, the ball doesn’t fall much faster. It has reached its terminal velocity, the speed at which the drag of air resistance balances the pull of gravity. At seven seconds it has fallen about 158 meters, just over 500 feet. In practice, it’s impossible to shoot a ball of this size and weight to 500 feet – trust me on this. I can show you how to prove it later. With a really strong lift and a three foot tube, you will be lucky to get it to 300 feet. Let’s say you shoot one, click the stopwatch when it is at the highest point and again when it hits the ground, and the time is 3.5 seconds. Read across from the 3.5 second line and find that the maximum height was about 53 meters, 174 feet. Easy.

I'm not able to upload the .XLS spreadsheet file, as it's not an allowed image or media file, but you can build it yourself in a few minutes. If you follow exactly the layout in the two pictures above, here are the formulae to put in the various boxes. Find the column and row (letter/number), then copy and paste each formula exactly as shown into the correct box. The equals sign and everything to the right of it is the formula. If there's no equals sign, just type in the number.

The first group are the example data - these are the boxes you can change with your real data. But if you don't put something in these boxes, it causes errors in some of the formulae later.

box B3 4.45 box B4 50 box B5 0 box B6 0 box B7 0.5 box B8 0.3

Everything else takes part in the calculation.

box D3 =B3*B3*3.14159/40000 box D4 =B4/1000

box A12 0 box B12 =B5 box C12 =B6 box D12 =C12*3.2808 box E12 9.81

box A13 =A12+$B$7 box B13 =B12+E12*$B$7 box C13 =C12+B12*$B$7+0.5*E12*$B$7*$B$7 box D13 =C13*3.2808 box E13 =9.81-($B$8*$D$3*B13*B13/$D$4)

You don't actually see those formulae in the boxes. As soon as you click or press Enter, the formula is calculated and the result, a number, shows up in the box. But if you click on a box with a formula, you can see it in the editing box at the top of the screen.

Now, highlight the first five boxes (A..E) on line 13. To do this, click on box A13 and, while holding the left mouse button down, drag the cursor across to E13, then let go of the button. The five boxes should become highlighted in a different color from the rest.

On the keyboard, press control-C to copy these boxes. In Microsoft Excel, they will become outlined. In Open Office, nothing seemed to happen, but that's ok.

Next, click on box A14 and, while holding the left mouse button down, drag the cursor across and down to box E52 (column E, row 52). That may involve dragging the cursor off the bottom of the screen, which will scroll up. When you get the cursor on E52, let go of the button. It doesn't have to be row 52, but it does have to be in column E. The entire block of boxes you just dragged through will be highlighted and turn a different color.

On the keyboard, press control-V to paste. All the highlighted boxes get the correct formula pasted into them automatically. You won't see the formula, though!

Congratulations, you are finished! The spreadsheet should have calculated the same numbers as the example above (though perhaps with different numbers of decimal places). If not, check that you copied the formulae into the correct boxes. Note, don't worry if boxes D3 and D4 aren't the same. The calculated values here are very small and you may not have enough decimal places turned on to display them properly. The numbers are still there, though, and the right values get used in all the calculations.

SAVE THE FILE.

Peret (Pete Hand)

**Easy shell height calculator**

As a natural progression to the shell height calculator above, it would make it easier to calculate the height given the total flight time as it can be difficult to judge the apogee of a shell in flight. It is clearly easier to time the flight from firing to impact.

This is a little more difficult to do as the drag coefficient is dependent on the velocity and the shape of the projectile. However the drag coefficient of a sphere at varying speeds is well known. Charts of the coefficient vs Reynolds number are available. So the program I have developed takes into account the speed of the shell to read the drag coefficient from the included table.

Screen shot of the calculator

There are only three things that need to be entered into the program, shell mass, actual shell diameter and flight time (In the green boxes). Once this is done you then hit the calculate button to work out the maximum height along with muzzle velocity and rise time.

The rise time is useful as it gives the correct fuse length for the shell so that it breaks at apogee. It also helps you to tune your lift to that of commercial items. You may want the fuse time to be for instance 3 seconds on a three inch shell, so by changing the flight time you can work out the total flight time required for the shell to apogee at in that time. Then you can tune your lift on a dummy to get the flight time the same as the calculated.

There are a number of other items that you can change on the sheet. The items in light blue affect the accuracy of the calculation and the time it takes the program to converge on the answer.

If you want to get very advanced then you can increase the number of points in the Reynold's number table to increase the accuracy of the drag calculations. However if you are not sure what you are doing with this it could lead to some very poor results.

So click here to get your copy of the spreadsheet. height calculator

Digger