The total energy of the system is 

but due to the face that the mass currently has no velocity, the kinetic energy term goes to zero.

Plugging in the given values, we can solve for the total energy of the system:

Relevant equations:

Write expressions for the initial kinetic and potential energies, if the spring is initially stretched to the maximum amplitude before being released.

Write expressions for the final kinetic and potential energies when the spring crosses the equilibrium point.

Use conservation of energy to equate the initial and final energy sums.

Solve the equation to isolate .

The units for angular frequency, , are radians per second.

We need to derive the equation for angular frequency using conservation of energy.

Rearrange to solve for the velocity:

The velocity is also the product of angular frequency and the distance of the oscillation:

Use this equation to derive the equation for angular frequency:

Finally, use our given mass and spring constant to solve:

At the equilibrium position, the spring does not contribute any potential energy. It is neither stretched, nor compressed.

All of the energy in the system is kinetic energy, resulting from the given velocity:

To solve this question, we will need to use conservation of energy. With no displacement, the ball has a velocity of  and zero displacement. At its maximum displacement, the velocity will be zero and all the energy will be converted to spring potential energy.

Use our given values to solve:

When a mass is in free fall, its potential energy  is increasing. In this instance, all of that energy must be counteracted by the bungee cord, which we can treat like a spring , for the person to come to a stop. This means that we can set these two equations as equal to one another:

We can then rearrange this equation to solve for , the mass at which the bungee cord can be stretched as far as the ground:

Substituting in the known values, we can solve for :

All the energy that is stored up by the compressed spring needs to turn into gravitational potential energy for the stuntman to come to a rest 7.0 m in the air. Gravitational potential energy is modeled by the equation   and the energy in the spring system is modeled by the equation .  We can thus set these equations equal to one another:

We then rearrange the resulting equation for the compressed distance, which is :

Substituting in the given values, we can solve for :

We start with , where  is the point the mass is traveling to,  is the maximum displacement,  is , and  is the time it takes to travel the distance.

Rearranging this equation for  gives

Here,  is  because the point we're looking for is on the other side of the equilibrium point.

We compare the force from Hooke's Law to the force from gravity and solve for displacement:

Rearranging this equation to solve for , we get

Plugging in the given values, we get

Solving for , we get

When the spring is at its maximum compression, its potential energy is maximized and its kinetic energy is 0. When the mass is at the spring's equilibrium point, all of the potential energy is converted into kinetic energy, so kinetic energy is maximized (and equal to the maximum potential energy) and potential energy is 0. We set these two equations equal to each other and solve for velocity.