Final answer:
In Physics, the energy stored in a spring and hence the distance it must be compressed to shoot a projectile at a higher speed is determined by Hooke's Law. If the speed is quadrupled, the spring must be compressed to a distance that allows it to store 16 times the original energy.
Step-by-step explanation:
The subject of this question is Physics, particularly focusing on the concepts of work, energy, and springs (Hooke's Law). When a spring gun shoots out a plastic ball at an initial speed v0 and then at a speed increased by a factor of 4, the energy delivered to the ball from the spring is also increased by a factor of 16 (since kinetic energy is proportional to the square of the speed). Hooke's Law states that the force needed to compress a spring is directly proportional to the displacement, and the energy stored in a spring is ½ k x^2 where k is the spring constant and x is the compression distance. Therefore, if the speed is increased by a factor of 4, the spring needs to be compressed more to store more energy. To find the required compression distance, we would use the relationship that the energy stored (and thus the compression distance squared) is proportional to the square of the final speed.