Final answer:
To compress a spring twice as far in a toy dart gun requires four times the work. If the lower setting takes 5.0 J, the higher setting requires 20 J since work done is proportional to the square of the displacement.
Step-by-step explanation:
The question is about the work done to compress the spring in a toy dart gun. When compressing a spring, the work done is directly related to the spring's constant and the displacement squared, following Hooke's Law. For a spring compressed to a certain distance, if it takes 5.0 J of work, compressing it twice as far would require four times the work since the work done is proportional to the square of the displacement (Work ≈ Displacement²).
Therefore, if the lower setting requires 5 J, the higher setting with double the compression distance requires 4 x 5 J = 20 J of work. This is because potential energy in a spring is given by ½ k x², where k is the spring constant and x is the displacement. Doubling the displacement (x) would quadruple the potential energy (since x is squared), hence quadrupling the work needed.