Science Help Needed!!!?

Question

I am trying to create a device that can launch a ball +/-0.5m away from a line 4m away from the device. I have already got the blueprint ready and the device I created is a mini launcher, something like: _
____________ | |
|Ball is here-----|-------| |
|___________| |_ |

Basically, I need to launch a ball.

Explaining the blueprint:
The thing that holds the ball is a container.
The dotted line is a toilet roll with tissue paper.
The thing on the right side of the toilet roll is a handle, which will supposedly be tied with 2 rubberbands (making a '+' sign) and when I pull it backwards and let go, the toilet roll will shoot out the ball and the launching handle will not be able to go through because of its size so the device works very well.

Now coming to the question:
WHAT THEORY OF SCIENCE DOES THE SHOOTING OF THE BALL RELATE TO?

(1st appropriate one WITH sources gets 10 pts)

The actual blueprint: __
____________ | |
|Ball is here-----|-------| |
|___________| |_ |

I have less than 12 hours!!!!

Answer 1

You are converting Potential energy into Kinetic Energy.

Potential energy stored in your elastic bands transfers the energy into the ball which then has Kinetic Energy as it flies through the air (similar to that of a bow and arrow.)

See Link-:
http://www.glenbrook.k12.il.us/gbssci/phys/Class/energy/u5l1b.html#elastic

Remember Kinetic Energy = 1/2 * mass * velocity^2
Elastic Potential = Force = k * x (where k is the spring constant and x is the distance it is pulled from its equilibrium position.)

You can convert the two using F = mass * acceleration (and re-arranging for mass, then substitute it into the Kinetic Energy Equation.)

Answer 2

How about F=ma When you apply a force to a mass you accelerate it. The force is equal to the mass * its acceleration

Answer 3

Use E = mc^2 which is energy to explain your work.