To determine the minimum energy stored in the spring when the dart gun was loaded, we need to use the conservation of energy principle, which states that the initial energy of a system is equal to its final energy. In this case, the initial energy of the system is the potential energy stored in the spring, while the final energy of the system is the kinetic energy of the dart when it reaches its maximum height.
The potential energy stored in the spring is given by:
PE = (1/2)kx^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
We don't know the values of k and x, so we need to find them using the given information. The displacement of the spring is equal to the height the dart is lifted, which is 50 cm or 0.5 m. The spring constant is related to the force required to compress or stretch the spring by a certain amount. We can use Hooke's law to find the spring constant:
F = kx
where F is the force required to compress or stretch the spring by x.
The force required to launch the dart vertically is equal to the weight of the dart:
F = mg
where m is the mass of the dart and g is the acceleration due to gravity (9.81 m/s^2).
Substituting the values, we get:
kx = mg
k = mg/x
k = (0.025 kg)(9.81 m/s^2)/(0.5 m)
k = 0.4905 N/m
Now we can calculate the minimum potential energy stored in the spring:
PE = (1/2)kx^2
PE = (1/2)(0.4905 N/m)(0.5 m)^2
PE = 0.0613 J
Therefore, the minimum energy stored in the spring when the dart gun was loaded is 0.0613 J.