Asolid block, with a mass of 8.58kg, on a frictionless surface is pushed directly onto a horizontal spring, with a spring constant value of 368n/m, until the spring is compressed by 11cm. the block is then released. what is the block's final speed, in m/s?

In this problem, we are talking about Mechanical Energy () which is the addition of the Kinetic Energy (energy of the body in motion) and Potential Energy (It can be Gravitational Potential Energy or Elastic Potential Energy, in this case is the second):

  (1)

The Kinetic Energy is:

Where is the mass of the body and its velocity

And the Potential Energy (Elastic) is:

Where is the spring constant and is the the position of the body.

Knowing this, the equation for the Mechanical Energy in this case is:

  (2)

Now, according to the Conservation of the Energy Principle, and knowing there is not friction, the initial energy must be equal to the final energy :

   (3)

   (4)

At the beginning, the block has a , because it starts from rest, this means the initial energy is only the Potential Elastic Energy:

   (5)

After the spring is compressed, is in its equilibrium point X=0, so . Then the block is released. This means the final energy is only the Kinetic Energy

   (6)

Now, we have to substitute (5) and (6) in (3):

Finally: