The formula for potential energy is .

Given the values for the mass, height, and gravity, we can solve using multiplication. Note that the height is negative because the book falls in the downward direction.

The formula for potential energy is .

Given the values for the mass, height, and gravity, we can solve using multiplication. Note that the height is negative because the book falls in the downward direction.

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.

Known

Unknown

When something does work on an object it changes the total energy of the object.  In this case, the work done by the person converts to kinetic energy as the stone is launched.  This kinetic energy is then turned into gravitational potential energy when the stone is at the highest point of the peak.

Work done = Kinetic Energy when the stone leaves the hand = Gravitational Potential Energy at the peak

We can set the work done to the gravitational potential energy

Plug in our known values and solve.

Known

We need to convert our velocity to 

Unkonwn

The easiest way to solve this problem is through conservation of energy.  When the car is rolling along the road it has kinetic energy.  Once the spring brings it to a complete stop, the car has elastic potential energy. According to the law of conservation of energy, these two values should be equal to one another

We can now plug in our known values and solve for the missing variable