Final answer:
To make the dart travel 4 times farther, the spring should be compressed to store 4 times the energy, so the answer is 20 J for the higher setting as work required is quadratic to the compression distance.
Step-by-step explanation:
The student has asked about the relationship between the compression of a spring and the distance a dart fired from a spring-powered dart-gun will travel. According to Hooke's Law, the force exerted by a spring is directly proportional to the distance it is compressed. The work done to compress a spring, which is a form of potential energy, is proportional to the square of the compression distance. Therefore, to fire a dart 4 meters, compared to 1 meter, the spring must be compressed a distance that will quadruple the energy stored.
The given problem states that compressing the dart gun to a lower setting takes 5.0 J of work. If the spring is compressed twice as far for the more powerful setting, the work done on the spring would be 4 times as much due to the spring compression work relationship being quadratic. Therefore, the amount of work required for the higher setting would be 20 J.