Final answer:
The maximum speed of the mass in the spring-mass system is found to be 3m/s, using the conservation of mechanical energy. The sum of initial potential and kinetic energies is equal to the kinetic energy at the point of maximum speed.
Step-by-step explanation:
To find the maximum speed during the motion of the spring-mass system, we can use the conservation of mechanical energy principle. Initially, the system has both potential energy (due to spring compression) and kinetic energy (due to the mass's velocity). As the spring releases, this potential energy is converted into kinetic energy.
The potential energy in the spring when compressed is calculated using the formula PEs = ½ k x2, where k is the spring stiffness, and x is the displacement of the spring from its equilibrium position. Likewise, the initial kinetic energy of the mass is given by KEi = ½ m v2, where m is the mass and v is the initial speed.
Conservation of energy states that the total mechanical energy at the beginning will be equal to the total mechanical energy at any other point in the motion, assuming no energy loss to friction. During the motion when the spring is at its equilibrium position (not compressed or stretched), all the potential energy would have been converted to kinetic energy. Therefore, we have:
PEs + KEi = KEmax
Substituting the given values:
½ × 200N/m × (0.1m)2 + ½ × 0.4kg × (3m/s)2 = ½ × 0.4kg × (vmax)2
Solving for vmax, we get the maximum speed the mass will have during its motion, which is 3m/s. Thus, the correct answer is C. 3m/s.