1. 0.42 Hz
The frequency of a simple harmonic motion for a spring is given by:

where
k = 7 N/m is the spring constant
m = 1 kg is the mass attached to the spring
Substituting these numbers into the formula, we find

2. 2.38 s
The period of the harmonic motion is equal to the reciprocal of the frequency:

where f = 0.42 Hz is the frequency. Substituting into the formula, we find

3. 0.4 m
The amplitude in a simple harmonic motion corresponds to the maximum displacement of the mass-spring system. In this case, the mass is initially displaced by 0.4 m: this means that during its oscillation later, the displacement cannot be larger than this value (otherwise energy conservation would be violated). Therefore, this represents the maximum displacement of the mass-spring system, so it corresponds to the amplitude.
4. 0.19 m
We can solve this part of the problem by using the law of conservation of energy. In fact:
- When the mass is released from equilibrium position, the compression/stretching of the spring is zero:
, so the elastic potential energy is zero, and all the mechanical energy of the system is just equal to the kinetic energy of the mass:

where m = 1 kg and v = 0.5 m/s is the initial velocity of the mass
- When the spring reaches the maximum compression/stretching (x=A=amplitude), the velocity of the system is zero, so the kinetic energy is zero, and all the mechanical energy is just elastic potential energy:

Since the total energy must be conserved, we have:

5. Amplitude of the motion: 0.44 m
We can use again the law of conservation of energy.
-
is the initial mechanical energy of the system, with
being the initial displacement of the mass and
being the initial velocity
is the mechanical energy of the system when x=A (maximum displacement)
Equalizing the two expressions, we can solve to find A, the amplitude:

6. Maximum velocity: 1.17 m/s
We can use again the law of conservation of energy.
-
is the initial mechanical energy of the system, with
being the initial displacement of the mass and
being the initial velocity
is the mechanical energy of the system when x=0, which is when the system has maximum velocity, 
Equalizing the two expressions, we can solve to find
, the maximum velocity:
