Final answer:
The minimum energy stored in the spring when the dart gun was loaded and the dart reached a maximum height of 9 m is 8.82 Joules, calculated using the potential energy formula Potential Energy = m × g × h.
Step-by-step explanation:
To determine the minimum energy stored in the spring of the dart gun when it was loaded, we must consider the conservation of mechanical energy, which states that the total mechanical energy (the sum of potential energy and kinetic energy) in a system remains constant if only conservative forces are doing work.
At the point of maximum height, the dart comes to a temporary stop before falling back down, which means its kinetic energy is zero at that point. Therefore, all the energy in the system is potential energy given by the formula Potential Energy = m × g × h. To find the minimum energy stored in the spring, we plug in the values: mass (m) = 0.1 kg, acceleration due to gravity (g) = 9.8 m/s², and height (h) = 9 m.
Using the formula, we calculate the minimum energy stored in the spring as follows: Potential Energy = 0.1 kg × 9.8 m/s² × 9 m = 8.82 J. This represents the energy converted from spring potential energy to gravitational potential energy at the peak of the dart's flight, neglecting any air resistance.