Why does kinetic energy equal potential energy

The fundamental laws of conservation are;. Returning to our example above, the 'conservation of socks' is, in fact, a consequence of the law of conservation of mass. It should be noted that in the context of nuclear reactions, energy can be converted to mass and vice-versa. In such reactions, the total amount of mass plus energy doesn't change. Therefore the first two of these conservation laws are often treated as a single law of conservation of mass-energy.

Within a closed system, the total amount of energy is always conserved. This translates as the sum of the n changes in energy totaling to 0. An example of such a change in energy is dropping a ball from a distance above the ground. The energy of the ball changes from potential energy to kinetic energy as it falls. Because this is the only change in energy within our system, we will take a simple physical problem and model it in order to demonstrate.

An object of mass 10kg is dropped from a height of 3m. What is its velocity when it is 1m above the ground? The Potential Energy of the object at a height of 1m above the ground is given in a similar fashion.

By definition, the change in Potential Energy is equivalent to the change in Kinetic Energy. The initial KE of the object is 0, because it is at rest. Hence the final Kinetic Energy is equal to the change in KE. This follows because we can actually use the equations for energy to generate the above kinematic equation. From Wikibooks, open books for an open world.

If the kinetic energy increases, the potential energy decreases, and vice-versa. In this case the following rule applies: The total energy of a system kinetic plus potential increases by the amount of work done on the system, and decreases by the amount of work the system does.

This leads us to consider the conservation of energy and other quantities. In many cases, "you get out what you put in". The fundamental laws of conservation are; conservation of mass conservation of energy conservation of momentum conservation of angular momentum conservation of charge Returning to our example above, the 'conservation of socks' is, in fact, a consequence of the law of conservation of mass.

Therefore the first two of these conservation laws are often treated as a single law of conservation of mass-energy Combining these laws with Newton's laws gives other derived conserved quantities such as conservation of angular momentum Within a closed system, the total amount of energy is always conserved.

It is important to understand that each form of energy does not exist separately but undergoes change from one to another without a net loss in energy. Consider this example:. As a projectile is launched into the air KE is at its maximum.

As the projectile gains altitude PE becomes greater than KE. At the top of its arc, PE is at its maximum. The whole cycle reverses itself on the way down. As you can see, the loss in KE is a gain in PE. Energy is not lost but conserved. Energy cannot be created nor destroyed. It can be transformed from one form to another, but the total amount of energy never changes.

Therefore, we consider this system to be a group of single-particle systems, subject to the uniform gravitational force of Earth. By definition, this work is the negative of the difference in the gravitational potential energy, so that difference is. Its Native American name, Massachusett , was adopted by settlers for naming the Bay Colony and state near its location. A kg hiker ascends from the base to the summit.

What is the gravitational potential energy of the hiker-Earth system with respect to zero gravitational potential energy at base height, when the hiker is a at the base of the hill, b at the summit, and c at sea level, afterward?

The altitudes of the three levels are indicated. First, we need to pick an origin for the y -axis and then determine the value of the constant that makes the potential energy zero at the height of the base. Then, we can determine the potential energies from Figure , based on the relationship between the zero potential energy height and the height at which the hiker is located. Besides illustrating the use of Figure and Figure , the values of gravitational potential energy we found are reasonable.

The gravitational potential energy is higher at the summit than at the base, and lower at sea level than at the base. Gravity does work on you on your way up, too! It does negative work and not quite as much in magnitude , as your muscles do.

But it certainly does work. Similarly, your muscles do work on your way down, as negative work. The numerical values of the potential energies depend on the choice of zero of potential energy, but the physically meaningful differences of potential energy do not.

Check Your Understanding What are the values of the gravitational potential energy of the hiker at the base, summit, and sea level, with respect to a sea-level zero of potential energy?

In Work , we saw that the work done by a perfectly elastic spring, in one dimension, depends only on the spring constant and the squares of the displacements from the unstretched position, as given in Figure. Therefore, we can define the difference of elastic potential energy for a spring force as the negative of the work done by the spring force in this equation, before we consider systems that embody this type of force. The potential energy function corresponding to this difference is.

Then, the constant is Figure is zero. Other choices may be more convenient if other forces are acting. When the spring is at its unstretched length, it contributes nothing to the potential energy of the system, so we can use Figure with the constant equal to zero.

The value of x is the length minus the unstretched length. Calculating the elastic potential energy and potential energy differences from Figure involves solving for the potential energies based on the given lengths of the spring. When the length of the spring in Figure changes from an initial value of Using 0.

A simple system embodying both gravitational and elastic types of potential energy is a one-dimensional, vertical mass-spring system. This consists of a massive particle or block , hung from one end of a perfectly elastic, massless spring, the other end of which is fixed, as illustrated in Figure.

Assuming the spring is massless, the system of the block and Earth gains and loses potential energy. We need to define the constant in the potential energy function of Figure.

Note that this choice is arbitrary, and the problem can be solved correctly even if another choice is picked. We must also define the elastic potential energy of the system and the corresponding constant, as detailed in Figure.

The equilibrium location is the most suitable mathematically to choose for where the potential energy of the spring is zero. Therefore, based on this convention, each potential energy and kinetic energy can be written out for three critical points of the system: 1 the lowest pulled point, 2 the equilibrium position of the spring, and 3 the highest point achieved. We note that the total energy of the system is conserved, so any total energy in this chart could be matched up to solve for an unknown quantity.

The results are shown in Figure. The block is pulled down an additional [latex] 5. In parts a and b , we want to find a difference in potential energy, so we can use Figure and Figure , respectively. Each of these expressions takes into consideration the change in the energy relative to another position, further emphasizing that potential energy is calculated with a reference or second point in mind. By choosing the conventions of the lowest point in the diagram where the gravitational potential energy is zero and the equilibrium position of the spring where the elastic potential energy is zero, these differences in energies can now be calculated.

In part c , we take a look at the differences between the two potential energies. The difference between the two results in kinetic energy, since there is no friction or drag in this system that can take energy from the system.

Even though the potential energies are relative to a chosen zero location, the solutions to this problem would be the same if the zero energy points were chosen at different locations. Suppose the mass in Figure is in equilibrium, and you pull it down another 3. Does the total potential energy increase, decrease, or remain the same?