Final answer:
The distance a ball travels when launched from a spring follows the principles of energy conservation, with potential energy being converted into kinetic energy. By using the potential energy formula, the range can be calculated from the spring constant and the displacement upon release.
Step-by-step explanation:
To determine how far a ball will go when launched from a compressed spring, we need to understand the physics of springs and energy conversion. The distance a ball travels when launched from a spring is calculated by energy conservation principles. The potential energy stored in a compressed spring is converted into kinetic energy when the spring is released.
If the relationship between spring compression and height is quadratic, then the equation that would likely be used is h = kx^2, where h is the height, x is the compression, and k is a constant that relates the two. With a spring compression of 3 cm (0.03 m), we can use the spring constant and the potential energy formula, U = 1/2kx^2, where k is the spring constant and x is the displacement from the equilibrium position.
Considering the spring constant k and displacement x, the potential energy U can be plugged into the quadratic equation to solve for the height and, subsequently, the range.
As an example, if we calculate the potential energy with a spring constant of 5000 N/m and a displacement of 0.03 m:
U = 1/2 * 5000 N/m * (0.03 m)^2 = 2.25 J
This energy is then converted into kinetic energy at the point of launch, which is transferred into both vertical and horizontal motion. The specifics of the trajectory would depend on the angle of launch, but with maximum compression, one would expect maximum range under ideal conditions.