Using the conservation of energy principle, the speed of the box at the moment when the elastic potential energy is fully converted to kinetic energy is approximately 6.057 m/s.
To calculate the speed of the box when the elastic potential energy is fully converted to kinetic energy, we use the conservation of energy principle. The potential energy stored in the spring when it is extended is given by the formula:
U = 1/2 k x^2
where U is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
When the box is released, this potential energy is converted into kinetic energy, which is given by:
KE = 1/2 m v^2
where KE is the kinetic energy, m is the mass of the box, and v is the velocity.
Assuming all potential energy is converted to kinetic energy, we can set U equal to KE and solve for v:
1/2 k x^2 = 1/2 m v^2
Solving for v, we get:
v = √((k x^2) / m)
Plugging in the given values, k = 106 N/m, x = 0.15 m (since 15 cm = 0.15 m), and m = 0.065 kg (since 65 g = 0.065 kg), we get:
v = √((106 N/m × (0.15 m)^2) / 0.065 kg)
v ≈ √((106 × 0.0225) / 0.065)
v ≈ √(2.385 / 0.065)
v ≈ √(36.6923)
v ≈ 6.057 m/s
Therefore, the speed of the box at the moment when the elastic potential energy is fully converted to kinetic energy is approximately 6.057 m/s.