A 4.00 kg mass is connected to a spring with a spring constant of 9.00 N/m. If the initial velocity is 12.0 cm/s and the initial displacement is 4.00 cm, then what is the maximum velocity of the mass?

To solve this problem, we need to use the principle of conservation of energy. The total energy in a simple harmonic motion is constant and is the sum of kinetic and potential energy.

1. First, we need to convert the initial velocity from cm/s to m/s by dividing by 100. So, the initial velocity (v_i) is 0.12 m/s.

2. The initial kinetic energy (KE_i) can be calculated using the formula KE = 1/2 * m * v^2, where m is the mass and v is the velocity. Substituting the given values, we get KE_i = 1/2 * 4.00 kg * (0.12 m/s)^2 = 0.0288 J.

3. The initial potential energy (PE_i) can be calculated using the formula PE = 1/2 * k * x^2, where k is the spring constant and x is the displacement. Substituting the given values, we get PE_i = 1/2 * 9.00 N/m * (4.00 cm)^2 = 0.072 J. Note that we need to convert the displacement from cm to m by dividing by 100.

4. The total initial energy (E_i) is the sum of the initial kinetic and potential energy. So, E_i = KE_i + PE_i = 0.0288 J + 0.072 J = 0.1008 J.

5. At the point of maximum velocity, the mass is at the equilibrium position, so the potential energy is zero. Therefore, the total energy is equal to the kinetic energy.

6. We can find the maximum velocity (v_max) by rearranging the kinetic energy formula to solve for v: v = sqrt((2KE)/m). Substituting the total energy for KE, we get v_max = sqrt((20.1008 J)/4.00 kg) = 0.224 m/s.

So, the maximum velocity of the mass is 0.224 m/s.