Is The Sum Of Kinetic And Potential Energy Always Positive?
Kinetic energy is the energy associated with motion. An object that is moving has kinetic energy. The amount of kinetic energy depends on the object’s mass and velocity. The greater the mass and velocity, the more kinetic energy the object has. Potential energy is stored energy based on an object’s position or arrangement. There are different types of potential energy like gravitational potential energy and elastic potential energy.
The main question we want to address is whether the sum of an object’s kinetic and potential energies is always positive. In other words, can the total amount of energy be negative? We will examine the principles of conservation of energy and situations where the total energy may be positive, negative, or zero.
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. It is directly proportional to the object’s mass and the square of its velocity. The faster an object moves, the more kinetic energy it possesses. Some examples of kinetic energy include:

A moving car, truck, train, or airplane

A flowing river or stream

Wind

A baseball being thrown or hit

A person running, walking, or dancing
Kinetic energy increases as an object speeds up and decreases as it slows down. The formula for kinetic energy is:
KE = 1/2 mv^{2}
Where m is the object’s mass and v is its velocity. Kinetic energy is directly proportional to both of these factors.
Potential Energy
Potential energy is the stored energy an object has due to its position or state. There are several types of potential energy:
 Gravitational Potential Energy – This is energy stored due to an object’s height above the ground in a gravitational field. For example, a book held above the ground has gravitational potential energy that can be converted to kinetic energy if it falls. The higher the book, the more potential energy it has.
 Elastic Potential Energy – This is energy stored in elastic materials that are deformed. For example, a stretched rubber band has elastic potential energy that can be converted to kinetic energy if released. The more the band is stretched, the more potential energy it has.
 Chemical Potential Energy – This is energy stored in the bonds between atoms and molecules. This energy can be released in chemical reactions. For example, the bonds in fuel molecules like gasoline contain chemical potential energy that is released as heat when the fuel burns.
In general, potential energy depends on the position, shape, or chemical composition of an object. This stored energy can later be converted to kinetic energy when the object moves.
Conservation of Energy
The principle of conservation of energy states that the total energy in an isolated system remains constant. Energy cannot be created or destroyed, only converted from one form to another. For example, when a ball falls, its potential energy is converted to kinetic energy as it accelerates due to gravity. The total amount of energy before and after the fall remains the same.
This principle applies to all isolated systems, where no energy can enter or leave. In practical systems there is often some exchange of energy with the surroundings, for example through friction, so the total energy goes down slightly. But in an ideal isolated system, the total energy remains fixed according to the conservation of energy principle.
This is an important physical law that allows us to analyze changes in energy during processes like falling objects, swinging pendulums, chemical reactions, and more. We can set up an energy balance equation where the total initial energy equals the total final energy, accounting for all transfers between potential, kinetic, thermal, and other energy types within the system.
Negative Potential Energy
Potential energy can be negative in certain situations. Here are some examples where potential energy is negative:
Gravitational Potential Energy
Gravitational potential energy depends on the height of an object relative to a zero point. If the object is positioned below the zero point, its gravitational potential energy will be negative.
For example, consider a ball held 1 meter below the ground. If we take the ground level as the zero point for gravitational potential energy, then the ball has a negative gravitational potential energy of mgh, where m is its mass, g is gravitational acceleration, and h is the height below the zero point.
Electric Potential Energy
The electric potential energy of a charged particle depends on its position relative to other charges. If the particle moves to a point where the electric potential is lower, its electric potential energy becomes negative.
For instance, when a positive charge moves farther away from a positive source charge, its electric potential energy decreases and can become negative as it gets sufficiently far from the source charge.
Chemical Potential Energy
In chemistry, the potential energy of a molecule or ion relative to some reference state is called its chemical potential. If the energy of the molecule/ion is lower than the reference state, its chemical potential is negative.
For example, gaseous water has a lower energy than liquid water. So water vapor has a negative chemical potential relative to liquid water at the same temperature and pressure.
Negative Kinetic Energy
Kinetic energy is defined as the energy of motion. The formula for kinetic energy is:
KE = 1/2mv^{2}
Where m is the mass of the object and v is its velocity. Kinetic energy is always considered a positive value in classical physics. This is because the square of the velocity, v^{2}, will always be positive regardless if the velocity is positive or negative.
However, in some modern extensions of physics like quantum field theory, it is possible to formally define negative kinetic energy. This occurs with certain types of hypothetical particles called tachyons that can travel faster than the speed of light and therefore have an imaginary (complex) mass.
Tachyons violate the theory of relativity and their existence is speculative. But in quantum field theory models allowing their existence, they can have negative kinetic energy values. This is because the square of an imaginary number is negative. So for tachyons, the value of v^{2} in the kinetic energy formula can be negative, leading to a negative kinetic energy.
But in most realworld scenarios described by classical mechanics and special relativity, kinetic energy is always a positive value. The velocity of ordinary particles like electrons, protons, atoms etc. cannot reach or exceed the speed of light, so their kinetic energy is always positive.
Sum of Energies
The sum of kinetic and potential energy in a closed system is generally positive, but there are some exceptions. According to the law of conservation of energy, the total energy in a closed system remains constant. Energy can transform between kinetic and potential, but the total amount of energy stays the same. Kinetic energy is the energy of motion and potential energy is stored energy due to an object’s position or configuration.
For example, a ball at the top of a hill has potential energy due to gravity. As it rolls down the hill, this potential energy gets converted into kinetic energy, the energy of motion. The sum of the potential energy at the start and the kinetic energy later remains constant, assuming no outside forces like friction act on the system. The potential energy decreases but the kinetic energy increases by the same amount.
So in most realworld scenarios, the sum of kinetic and potential energies in a system is positive. The total energy is conserved as one form changes into another. However, there are some special cases where the total energy or the potential energy value can be negative, which we’ll explore next.
Exceptions
While the sum of kinetic and potential energy in a closed system is usually positive, there are some exceptions where this is not the case:
If an object has a negative potential energy, such as when it is interacting through a repulsive force or is in an unstable high energy state, then the overall sum can be negative. For example, two positively charged particles that repel each other will have negative potential energy as they get closer together. If their kinetic energy is lower in magnitude than the potential energy, the overall sum will be negative.
Objects can also have negative kinetic energy in certain reference frames. For example, if an object is moving at a slower velocity than the reference frame itself, its kinetic energy will be negative relative to that frame. So in some cases, the object’s negative kinetic energy can make the overall sum with potential energy negative.
In quantum systems, the uncertainty principle means that particles can briefly borrow energy to come into existence as long as that energy is repaid quickly enough. So virtual particles can make the sum of energies negative for small amounts of time until the particles annihilate each other.
Additionally, in general relativity, the energy of an expanding space can be negative. When calculated relative to a flat spacetime, the kinetic and potential energy of cosmological expansion can sum to a negative value.
So while the total energy is conserved, there are certain exceptions where the sum of kinetic and potential energy specifically can be negative due to effects like negative potential energies, relative motion, quantum fluctuations, and cosmological expansion.
Summary
To restate the original question – Is the sum of kinetic and potential energy always positive? After reviewing kinetic energy, potential energy, conservation of energy, cases of negative potential energy, and negative kinetic energy, we can conclude that:
The sum of kinetic and potential energy is not always positive. While kinetic energy is always positive, potential energy can be negative in certain cases. Specifically, potential energy turns negative when the forces acting on an object tend to accelerate the object away from a reference point. For example, gravitational potential energy is negative when an object is below the zero level taken as a reference. Additionally, there are some exotic situations where kinetic energy can also go negative, such as particles moving faster than the speed of light in some cosmological theories.
However, the total mechanical energy, which is the sum of kinetic and potential energy, remains conserved according to the law of conservation of energy. The exceptions arise when considering potential energy relative to a chosen reference state. Overall, while kinetic energy is positive, potential energy can be negative, so their sum does not have to remain positive at all times.
Conclusion
In summary, while the sum of kinetic and potential energy is often positive, there are important exceptions to be aware of. Specifically, potential energy can be negative when the object is below the zero level for gravitational potential energy. Similarly, kinetic energy can be negative when an object is moving in the opposite direction of the positive coordinate axis.
When both potential and kinetic energy are negative, their sum will be negative as well. Additionally, the total mechanical energy of a system may be negative if there are nonconservative forces doing work, such as friction or air resistance.
Therefore, while the total energy is conserved according to the law of conservation of energy, the sum of kinetic and potential energy alone does not have to be positive in all cases. The key exceptions occur when either form of energy is negative based on the choice of reference level or coordinate axis orientation.