Final answer:
The speed of the box at the moment when the entire elastic potential energy is converted to the box's kinetic energy is approximately 5.25 m/s.
Step-by-step explanation:
To calculate the speed of a 61 g box attached to a horizontal spring when all its potential energy is converted to kinetic energy, we use conservation of energy principles. The potential energy stored in the spring is given by the formula U = ½ kx², where k is the spring constant and x is the extension of the spring. At the point where all the potential energy is converted into kinetic energy, the kinetic energy (K.E.) is equal to the potential energy (U).
We first convert the mass of the box to kilograms (m = 0.061 kg) and the extension to meters (x = 0.12 m). The spring constant is already in the correct unit (N/m).
Calculate the potential energy (U):
U = ½ (117 N/m) × (0.12 m)² = 0.84 J
Since the kinetic energy is equal to the potential energy at the point of complete conversion, we have K.E. = ½ mv² = U. Solving for v, the velocity, we get:
v = √(2U/m)
Substitute the values:
v = √(2 × 0.84 J / 0.061 kg)
Calculate the velocity:
v = √(27.54) = 5.25 m/s
The speed of the box at the moment when the elastic potential energy is fully converted to kinetic energy is approximately 5.25 m/s.