Final answer:
Using the conservation of energy, the maximum speed of each 1.80 kg mass when free of the spring is calculated using the spring constant (165 N/m) and the compression distance (0.20 m), resulting in a maximum speed of approximately 1.91 m/s.
Step-by-step explanation:
To find the maximum speed of each mass when it is no longer in contact with the spring, we can use conservation of energy. The potential energy stored in the spring when compressed is fully converted into the kinetic energy of the masses when they lose contact with the spring.
The formula for the spring's potential energy (PEspring) when compressed is given by PEspring = 1/2 k x2, where k is the spring constant and x is the displacement from its normal length. In this case, k = 1.65 N/cm (which we need to convert to N/m for consistency in units) and x = 20.0 cm.
To find speed, we set the spring's potential energy equal to the kinetic energy (KE) of one of the masses: 1/2 m v2 = 1/2 k x2. Solving for v, the speed, we use the expression v = sqrt(k/m) * x, where m is the mass of each object.
After converting k to N/m (1.65 N/cm = 165 N/m) and x to meters (20 cm = 0.20 m), we can compute the speed. The calculation yields:
v = sqrt((165 N/m) / (1.80 kg)) * 0.20 m = sqrt((165 / 1.80)) * 0.20 ≈ 1.91 m/s
Therefore, the maximum speed for each mass when it has moved free of the spring is approximately 1.91 meters per second.