In this problem, we are talking about Mechanical Energy ([tex]M[/tex]) which is the addition of the Kinetic Energy [tex]K[/tex] (energy of the body in motion) and Potential Energy [tex]P[/tex] (It can be Gravitational Potential Energy or Elastic Potential Energy, in this case is the second):
[tex]M=K+P[/tex] (1)
The Kinetic Energy is: [tex]K=\frac{1}{2}mV^{2}[/tex]
Where [tex]m[/tex] is the mass of the body and [tex]V[/tex] its velocity
And the Potential Energy (Elastic) is: [tex]P=\frac{1}{2}cX^{2}[/tex]
Where [tex]C[/tex] is the spring constant and [tex]X[/tex] is the the position of the body.
Knowing this, the equation for the Mechanical Energy in this case is:
[tex]M=\frac{1}{2}mV^{2}+\frac{1}{2}cX^{2}[/tex] (2)
Now, according to the Conservation of the Energy Principle, and knowing there is not friction, the initial energy [tex]M_{i}[/tex] must be equal to the final energy [tex]M_{f}[/tex]:
[tex]M_{i}=M_{f} [/tex] (3)
[tex]M_{i}=\frac{1}{2}m{V_{i}}^{2}+\frac{1}{2}c{X_{i}^{2}}[/tex] (4)
At the beginning, the block has a [tex]V_{i}=0[/tex], because it starts from rest, this means the initial energy [tex]M_{i}[/tex] is only the Potential Elastic Energy:
[tex]M_{i}=\frac{1}{2}c{X_{i}^{2}}[/tex] (5)
After the spring is compressed, is in its equilibrium point X=0, so [tex]X_{f}=0[/tex]. Then the block is released. This means the final energy [tex]M_{f}[/tex] is only the Kinetic Energy
[tex]M_{f}=\frac{1}{2}m{V_{f}}^{2}[/tex] (6)
Now, we have to substitute (5) and (6) in (3):
[tex]\frac{1}{2}c{X_{i}^{2}}=\frac{1}{2}m{V_{f}}^{2}[/tex]
[tex]V_{f}=\sqrt{\frac{c{X_{i}^{2}}}{m}}[/tex]
[tex]V_{f}= X_{i}\sqrt{\frac{c}{m}}[/tex]
[tex]V_{f}=0.11m\sqrt{\frac{368N/m}{8.58kg}}[/tex]
Finally:
[tex]V_{f}=0.7203m/s[/tex]
