Ballistic Pendulum Instructions

So you want to measure the muzzle velocity of a cannon.

The instructions here will let you figure it out.

To make it somewhat less hazardous we will be using a bean shooter as our cannon.

Print out the form you find here.  You will be filling in your data on the form and using the tables on it to calculate the projectile's velocity.
And print the large protractor quadrant you find here to use to measure the angle when you get to that part of the experiment.

You will need a projectile to shoot.  Navy beans work well.  Later you will need to know the weight of your projectile.  If your scale isn't sensitive enough to weigh one bean count out enough to weigh on your scale and then divide by the number of beans to get their average weight.  If you are selective and don't include very small or very large ones you will get a more accurate result.   Record the weight of one bean in the first box in the "Projectile mass" column.

Next the "cannon".  A straw that is large enough for the bean to slide through easily.  We found some very long ones that I am told you will get when you order a Bahama Mama in some establishments.  A wire through the straw near one end you will blow through will turn the straw into a muzzle loader.  That will make it less likely to accidentally fire the bean into your uvula (look it up).

Now we need something to catch the projectile.  We use a single sheet of paper rolled into a tube that is closed on one end.  Since the bean must stay in the tube we have found that it works better if you crumple the paper and flatten it some before you form the tube.  That way the bean is more likely to be caught and not bounce out.

We also need the weight of the pendulum, so you should weigh it now and record it on the data sheet in the first box in the column labeled "Pendulum mass".  You can divide and put the result in the "Ratio" column.

The catcher now becomes the bob of a pendulum.  By hanging it on three threads you can make sure it is level, in line with the cannon, and will swing in a straight line.  The tables that you will be using to determine the muzzle velocity of the cannon are based on a1.00 meter length of the pendulum, from the support point to the center of mass of the pendulum bob. 

Now you need a way to measure how much the pendulum is deflected when it catches the projectile.   Print out the protractor quadrant and fasten it so the circle on the quadrant is close to the pendulum support point for one of the threads and the 0 degree mark and the pendulum will swing in front of it. 

It is a good idea to color that thread so that it is easy to recognize when the pendulum is moving.

Here is a diagram showing how everything should look.

Now we are almost ready to shoot.  EVERYONE PUT ON YOUR SAFETY GLASSES.

Position your marksman far enough away so the air from the straw doesn't move the pendulum very much but close enough that she/he can hit the open end of the pendulum (at least sometimes).

Position the observer so he/she can see the thread in front of the quadrant with it in line with the 0 degree mark.

If everyone is ready, FIRE!

The observer should record in column titled "Measured angle A" the angle that the pendulums swings to when it catches the bean.

Repeat several times recording each result in the boxes in that column.

We now have everything we need to figure out the velocity of the projectile (bean).  Here is how it will work.  When the bean hits the pendulum its momentum is shared with the pendulum giving it some kinetic energy. The pendulum swings and because of that it rises.  So its kinetic energy is converted to potential energy.  It will stop when all of its kinetic energy is converted to potential energy.  That will happen at the greatest height (angle).  Since you recorded the angle you have a measure of the momentum and since you measured the weight you have everything you need to get the velocity of the bean.

Now go the the tables on the data sheet.   They have been set up so that the calculations outlined above are all taken care of for you.   First,  Look up (
on the table on the top right) how much the pendulum bob went up when it swung because the bean hit  and record the result next to the angle in the column labeled "Height from table on right".  If the angle you observed  is halfway between  the numbers on the table take the average of the heights from the  one above and one below it on the table.  

Go to the big table at the bottom of the page and find the column that has the height that most closely matches the height you got from the first table.  Then find the row that comes the closest to the ratio of the pendulum mass/projectile mass that you calculated earlier.    Where that row and column cross is the velocity of the projectile in meters per second.  If you want to know the result in miles per hour multiply the value in meters per second by 2.24.  If instead you want to get the result in feet per second multiply your meters per second answer by 3.28.

For a little more precision you can interpolate between the table values. If the number you have is half way between the numbers at the head of the columns you would use a number half way between the numbers in the table, if it was a third of the way from the smaller to the larger number at the head of the columns then the result would be found bu adding 1/3 of the difference to the smaller of the numbers in the columns you are using.

If you want you can repeat the experiment with other "ordnance".  We found that Nerf pistols work well.  Because their projectiles are a lot heavier they have more momentum than the beans even when going slower.  Therefore you will need a heavier pendulum so that it isn't knocked about too much.  Adding a few pennies to it may help.  You will want to make sure they stay attached when the dart hits the pendulum.  

You can just weigh the dart and pendulum, divide and enter the result, fire and record the angle and then use the tables again to determine the velocity of the projectile.

I would be interested in your results and if you come up with any other "
ordnance" that I could try. 

If you would like to do your own calculations for the velocity for other angles, pendulum lengths, or masses the equations are:

A = Maximum deflection angle

L= Length of the pendulum from the support point to its center of mass

H = 2 L (sin(A/2))2 = Calculated increase in height of the pendulum bob at the maximum deflection angle

g = acceleration of gravity in meters per sec2 =  9.8

Mpendulum = Mass of the pendulum

Mprojectile = Mass of the projectile

V = (1+ (M
pendulum/Mprojectile)) (2gH).5 = (1+ (Mpendulum/Mprojectile)) times the square root of (2gH) = Velocity of the projectile
Send me email with the speeds you or your class measures if you would like.

Have fun.

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