The correct answer is that the total energy at the top of the hill is equal to the total energy at the bottom of the hill. This is called the Law of Conservation of Energy. This law says that kinetic energy is equal to potential energy. We know that energy can not be created or destroyed, but that it can be transferred from one type to another. The total energy at the top of the hill is all potential energy. As the ball begins rolling down the hill, all the potential energy is converted to kinetic energy. Therefore, the total energy remains constant and energy is conserved.  

Determining expression for vertical component of ball's initial velocity:

Using conservation of energy:

Solving for 

At the maximum height, there will be no motion in the vertical direction, thus 

Plugging in values:

Using conservation of energy:

While at the maximum value of stretch (the amplitude), the velocity will be zero. While at the maximum velocity, the stretch will be zero.

Plugging in values:

Solving for 

At the maximum height, velocity is zero.

Solving for 

Plugging in values

Based on the information given, his throw would have started at above the ground. The pizza gained a maximum height of . Thus, the gravitational potential energy in relation to the starting position will be:

Since at it's maximum height, all of the energy will be gravitational, the work done on the pizza will be equal to the potential energy, thus:

Conservation of energy:

Solving for 

Plugging in values:

This question tests your understanding of conservation of energy, gravitational potential energy, and energy of spring compression. In this question, a  ball is dropped from an initial height of  from the ground, and then at the level of the ground, it contacts and compresses a spring, with all gravitational potential energy being transferred to the spring. You are then asked to calculate the spring constant. Because we are told that all gravitational potential energy is transferred to compressing the spring, we can set the gravitational potential energy equal to the spring compression energy, and solve for the spring constant as follows:

Next, we must isolate .

Therefore, the spring constant is .

This question tests your understanding of the concept of conservation of energy (energy can neither be created nor destroyed). In this example, initially all of the ball's energy is potential energy, as it is immobile at a height of  prior to being dropped. At the ball hits the ground, its energy is entirely transformed to kinetic energy. Thus, since we are assuming that no energy is lost to drag forces, the initial potential energy of the ball is equal to the final kinetic energy of the ball. We can solve for the ball's velocity as follows:

Potential Energy (initial) = Kinetic Energy (final)

Isolate the velocity term and solve.

Thus, the speed of the ball as it hits the ground is

This question tests your understanding of the concept of conservation of energy (energy can neither be created nor destroyed), and likewise of transitions between gravitational potential energy and kinetic energy. In this case, the  toy car starts with a potential energy of , since. Thus, as this is the maximal height of the car, and no forces other than gravity are acting on the car,  represents the maximal potential energy of the car.

We are then told that the car enters a loop with a radius of , indicating that the maximal height of the loop is also , and therefore, since no energy is lost to friction, the maximal potential energy of the car will still be at the peak of the ramp. We are explicitly asked to find the speed of the car when it is  off of the ground, on the loop.

There are a number of correct ways to calculate the speed of the car at this point. We will use the conservation of energy formula as follows:

Thus, the speed of the car at a height of  off of the ground on the loop is .

This problem tests your knowledge of conservation of energy, gravitational force, and spring force. First, we begin with a  block that is placed directly on a spring. We are then told that all of the gravitational energy on the block is transferred to the spring. Therefore, we can set the gravitational force on the block equal to the spring compression force as follows:

Next, we must isolate .

, and we can eliminate the negative sign since the question asks for the magnitude of the spring constant

Therefore, the spring constant is .