Originally Answered: Can anyone that is good in Physics solve this problem?
This is work/energy problem.
If you got a constant force on an object with a distance moved, you get work done right? Work, in SI units, is measured in jules. Likewise, energy is too. So what you have is a conservation of energy problem with a mass that had no initial velocity, a force appplied, then a mass with a velocity once it leaves the barrel with no additional force acting on it. Well, there is always gravity, but that is not important in this question.
To start, you have work done, W.
This work is equal to the kenitic energy of the mass when the mass leaves the barrel, so
Since you have the same work done on the larger mass, (2m), the second equation is
Now since these to equations both equal the same work, set them equal to eachother and solve for the second velocity in terms of the first.
To show the math, I will represent the velocity of the smaller mass with the letter Q.
the "halves" and the "m's" cancle out
Q^2 = 2(v^2)
Divide both sides by 2
(Q^2)/2 = v^2
Now that the square root of both sides
sqrt[(Q^2)/2] = v
Q/(2^(1/2)) = v
The reason why both cannonballs have the same amount of work done on them is because work only depends on is the force applied and the distance traveled of a object. Since the force is constant for any size projectile, as stated in the question, and the distance that the force is applied to the object, a.k.a the lenght of the barrel of the cannon, is constant, the work done will be the same.
Work = Force x Distance
("Mass" is not in the equation)
Since both the cannonballs have the same amount of work applied to them, they will have the same kinetic energy once they leave the barrel. The difference between them is their velocities.
Think about it, if you put the same amount of work, or energy, into throwing two different sized objects, the one with the smaller mass will go faster right? Think about the difference between a baseball and a bowling ball.