On a rotating platform with angular frequency \(\omega=1.5 \mathrm{rad} / \mathrm{s}\) a small spring cannon is fixed, at a distance \(R=2.0 \mathrm{~m}\) from the center. The small cannon, arranged with a vertical barrel, launches a projectile ( \(m=15 \mathrm{~g}\) ) by the release of a spring of elastic constant \(k=150 \mathrm{~N} / \mathrm{m}\), initially compressed by \(d=3.0 \mathrm{~cm}\). Neglecting the effect of gravity in the path of the projectile in the barrel of the gun, the size of the barrel and the compression length of the spring with respect to the distance along the vertical traveled by the projectile, calculate:

1. the speed of the projectile as it exits the cannon in the reference system integral with the platform;

2. the same speed observed in a fixed (inertial) reference frame;

3. the distance between the position of the cannon and the point where the projectile falls back on the platform. (Hint: assume for simplicity that the cannon fires when it is at the \((x, y)=(R, 0)\) coordinate point of the inertial frame).

4. The approximations to the dimensions of the small cannon given above are considered valid if they are less than \(10 \%\) of the dimensions corresponding to the object's trajectory, particularly the height attained. Are they therefore valid with the data provided?