The formula for gravitational potential energy is:

We are given the mass of the ball, height, and acceleration from gravity. Remember, since the displacement is downward it must be negative.

We need two formulas for this problem: the gravitational potential energy and the spring potential energy.

We know the weight of the box and the change in height when it is placed on the spring. These values allow us to calculate the change in potential energy. Due to conservation of energy, the total energy must remain constant. Initially, the box only has gravitational potential energy. In its final position, it has both gravitational and spring potential energy.

We know the initial height and final height of the spring. This also gives us the value of , the displacement of the spring.

We also know the weight of the box:

Now we can solve for the spring constant, .

Potential energy due to gravity is given by the equation:

We are given the mass of the branch and its height. Gravity is constant. Using these values, we can solve for the potential energy.

First, convert the mass of the branch to kilograms.

Then, use the equation to find the energy.

Potential gravitational energy is given by the equation:

We are told the height of the rock and its mass. Using the constant acceleration due to gravity, we can solve for the gravitational potential energy.

The formula for gravitational potential energy is . The only relevant variables are mass, gravity, and height. All of the information about velocity and angle are not needed to solve for the initial potential energy.

Plug in the given values for mass and height, and solve using the equation. Keep in mind that the height will be negative because the rock travels downward.

The formula for gravitational potential energy is:

We can solve for this value using the given mass of the rock, acceleration of gravity, and initial height.

This value is independent of the kinetic energy of the rock, and is not dependent on initial velocity.

Work is a vector quantity equal to the change in energy in a given direction. Our net work will be equal to product of force and displacement.

We can write this equation in terms of Newton's second law to incorporate our given values.

During its motion the weight travels a distance of upward, and then downward. Its total distance will be , however, we are looking for its displacement. The initial height and final height are equal, making the displacement equal to zero.

The total work will be zero because there was no net displacement.

The formula for work is , or work equals force times distance.

Since , you can expand the work formula a little bit to see .

The problem gives us the mass and the distance traveled, but not the acceleration. Given the acceleration, along with the mass and distance, we would be able to calculate the work.

The formula for work is , or work equals force times distance.

The problem gives us the mass, but is missing the distance. Given the distance, we would be able to calculate the work.

Work is a force times a distance or . 

Only one of these has a force times a distance: the weightlifter lifting a weight above his head.

The car moving at a constant velocity has no acceleration and therefore no force.

Typing essays does not require a force times a distance (though it sure feels like work!).

If the skier has no loss in energy, then no work was done, as work is also the measurement of the change in energy.

The man sitting in a chair has a constant velocity of zero, no acceleration, and therefore no force.