Here is a guide on how to make a marble cannon that obeys the laws of physics.
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Step 1: Acquiring the Materials
You will want to start off by going to your local hardware store and picking up a few items for the cannon. These will include
-4 Copper Splitrings
-2 Extended 7-16 in X 3 - 1/2 in springs
-2 pieces of plastic pipe (go for larger than you think you need because you can always cut it)
-Some wood of varying sizes (this will change based on how large you are making your cannon)
-A dowel rod to pus the marble out of your cannon
-And most importantly a lot of luck and patience.
Step 2: Building the Base
You will want to screw the wood you gathered together in a somewhat tall manner so that your cannon will be able to angle and adjust its shot. You will also want to add a piece of the pipe you got to your base so that you can adjust your cannons aim.
Step 3: The Firing Mechanism
Since you want your cannon to actually shoot the marble you will want to build a firing mechanism, we did this by attaching the 1/2 In copper splitrings to each other about an inch and a half apart, with one set on the pipe that we are using our barrel, and the other set on the dowel that we are using as our pushing force. From this, we attached the springs between the splitrings on both sides to give our cannon some power.
Step 4: Attaching the Firing Mechanism
Because we want to be able to actually shoot the cannon, we have to attach the firing mechanism to our base, this can be done rather simply by aligning the mechanism with the center of the pipe that you used earlier on the base, once you have this aligned correctly you will want to use something to keep it in place as holding it does not work too well (we tried). For this you can use your adhesive of choice, and if that fails, zip ties are a decent last-minute substitution.
Step 5: Calculations
Now that you have perfectly working marble cannon you will want to do some calculations in order to test its performance.
For this you will want to use the equation: t= 2 * the square root of "y" - "x" * tanØ, then divide all of that by g.
t= how long your marble was in the air
y= the negative of your launch height (from the top of your barrel to the floor)
x= how far your marble traveled
Ø= the angle of your barrel
g= the gravitational acceleration
Our calculations are shown in the image above and can be used as a rough guideline for how to do your equations.