To answer this question, we can address each type of energy separately. There is no conservation of energy in this problem; kinetic energy is not converted to potential energy as the man runs up the hill. Instead, he is accelerating, indicating an outside force that disallows conservation of energy.

First, we will find the maximum potential energy using the equation:

The man's mass and the acceleration of gravity will remain constant. The only changing variable is height. When the height is greatest, the potential energy will be the greatest. We can conclude that the potential energy will thus be greatest at the top of the hill.

Now we will look at the equation for kinetic energy:

The man's mass will remain constant, and the only changing variable will be the velocity. We are told that the man accelerates as he runs up the hill, indicating that his velocity is increasing. This tells us that he will reach a maximum velocity when he reaches the top of the hill, at which point he maintains a steady velocity along the top of the hill. Since kinetic energy is at a maximum when velocity is at a maximum, we can conclude that kinetic energy is greatest at the top of the hill.

For this problem, we're going to use the law of conservation of energy. Since we're looking for max velocity, we're going to say that the  of the system.

The formula for potential energy of a spring is 

Therefore:

Notice that the 's cancel out.

Plug in our given values.

Total mechanical energy is the sum of potential energy and mechanical energy.

We can expand this equation to include the formulas for kinetic and potential energy.

Since we are only looking at vertical energies, we need to find the initial vertical velocity to apply toward the kinetic energy.

To find the vertical velocity we use the equation .

We can plug in the given values for the angle and initial velocity to solve.

Now we have all the terms necessary to solve for the total energy. Keep in mind that the change in height is going to be negative, since the rock is traveling downward.

Mechanical energy is the sum of potential and kinetic energies.

Since we're only looking at the vertical components, we need to find the initial vertical velocity. This will be used for the vertical component of the kinetic energy. Use the sine function, initial velocity, and angle for this calculation.

Gravitational potential energy only exists in the vertical plane, so we do not need to manipulate the values. Use our vertical velocity, mass, and the height of the building to find the vertical mechanical energy at the point of release.

The formula for kinetic energy is:

Since the velocity we're working with is going to be in the vertical direction, we need to find the final . The best place to start is by finding the initial vertical velocity. To do that, we need to break the given velocity into its vertical component by using the sine function and the angle.

We know that the rock is going to travel a net of , as that's the displacement between the rock's initial position (on the building) and the ground. Using the appropriate motion equation, we can find the final velocity using the initial velocity, displacement, and acceleration.

Use this final vertical velocity and the mass of the rock to calculate the final kinetic energy in the vertical direction.

We can use potential energy to solve. Remember, your height and your gravity need to have the same sign, as they are moving in the same direction (downward). Either make them both negative, or use an absolute value.

Using conservation of energy, we know that . This tells us that the potential energy at the top of the hill is all converted to kinetic energy at the bottom of the hill. We can substitute the equations for potential energy and kinetic energy.

The masses cancel out.

Plug in the values, and solve for the velocity.

For this problem, the ball starts with both potential and kinetic energy. The point of maximum velocity will have no potential energy. We can solve setting the initial energy and final energy equal, due to conservation of energy.

The masses will cancel out from all of the terms.

Plug in the given values and solve for the final velocity. Remember, when the ball is on the ground it has a height of zero. 

The book initially has only potential energy. Right before it hits the ground, all the potential energy will be converted to kinetic energy. We can use the law of conservation of energy to set the initial and final energies equal.

Use the equations for potential energy and kinetic energy.

Now, plug in the values given to you in the problem and solve for the velocity.

Initially, the book has only potential energy. Right before it hits the ground, all the potential energy will have converted to kinetic energy. The two values will be equal based on the conservation of energy.

If we solve for potential, we can find kinetic energy.

Right before it hits the ground, the initial potential energy and the final kinetic energy will equal each other due to conservation of energy.

If we solve for initial potential, we can find final kinetic energy.

Plug in the values given. Remember that height is the change in height. Since the rock is headed downward, the height will be negative.

Multiply and solve.