# What does it physically mean for kinetic energy to be converted to gravitational potential energy (in the gravitational field)?

I can physically understand how kinetic energy can be converted to thermal energy. (vibration of particles)

But what's the equivalent understanding of throwing a ball up in the air, where the ball's kinetic energy decreases and it's gravitational potential energy increases because these two forms of energy are converted between each other. I understand that the gravitational field is involved in this process, where kinetic energy from the ball is being converted to the gravitation potential energy in the gravitational field. But how can I gain a physical understanding of this process?

When the ball's kinetic energy decreases and its gravitational potential energy increases, conversion is arguably a misleading word, suggesting as it does that one thing changes into another thing. Rather, energy is a conserved quantity, and one that is calculated according to a fixed recipe: for a ball near the Earth's surface (ignoring air resistance) we add $$mgh$$ to $$\frac12 m v^2.$$ The proportions of these two terms change as the ball rises, but we're still calculating the same thing – Energy.

Regarding your first sentence, what I understand is how a moving body, as it slows down, can make its own particles, and those of other bodies with which it is in contact, vibrate more vigorously. Again we find that the system's energy is conserved if we include an energy term (e.g. $$mc\Delta T$$), that deals with the vibrating particles.

This is an example of the situation that you have to assume the phenomenon as is in order to formulate a theory at all.

This is a 'choose your battles' type of situation. If you would insist on first developing a deeper understanding what gravitational potential energy is then you would only be bogging yourself down.

The physics community takes the known properties of gravitational interaction and proceeds from there.

As an example of what has been achieved by 'taking it from there':
The Einstein field equations are used to build computer models for the process of an astrophysical merger event: mergings of neutron stars and/or black holes. Data from the LIGO observatories are compared with a repository of profiles that have been generated using the computer models, and thus the researchers can infer from the measured profile what the respective masses were of the two objects that merged. This research has opened up a whole new area of astronomy.

I give the example of the interpretation of the LIGO observed events to emphasize that while the physics community may hit limits to how deep it can go, this doesn't necessarily put any limit on how far the science can reach.

I understand that the gravitational field is involved in this process, where kinetic energy from the ball is being converted to the gravitation potential energy in the gravitational field. But how can I gain a physical understanding of this process?

As @Philip Wood aptly pointed out, the word "conversion" is arguably a misleading term. In my view, "energy transfer" is more applicable in the case of the ball losing kinetic energy and gaining gravitational potential energy.

First, all types of energy (chemical, electrical, mechanical, thermal, nuclear, etc.) boil down to two basic forms: kinetic and/or potential energy. Kinetic energy is the the energy of motion. Potential energy is the energy of position.

Second, there are two basic mechanisms by which energy can be transferred from one thing to another: Heat and Work. Heat is energy transfer between things due solely to a temperature difference. Work is energy transfer due to force times displacement. The result of energy transfer is the increase and/or decrease in the kinetic or potential energy of the objects between which the transfer occurs.

In the case of the rising ball, the kinetic energy of the ball is transferred to the ball-earth system in the form of gravitational potential energy due to negative work done by the gravitational field. Gravity's work is negative because the force of gravity is opposite to the displacement of the object.

I can physically understand how kinetic energy can be converted to thermal energy. (vibration of particles)

Your understanding is correct. An example is an object sliding on a surface with friction that brings the object to a stop. Like gravity, in the case of the rising ball, friction does negative work (its force is opposite to the movement of the object). But in the case of friction, its negative work takes kinetic energy away from the object and transfers it to the contacting surfaces in the form of kinetic energy at the microscopic level. Gravity, on the other hand, transfers the energy to the ball-earth system in the form of gravitational potential energy. In either case, energy transfer occurs due to work.

Hope this helps.

Work is a force acting through a distance, and the work from the net force equals the change in kinetic energy. For a conservative force, by definition the work done by that force is independent of the path for the distance traveled, and the work done by a conservative force can be evaluated as the negative of the change in a potential energy for that force. So for a conservative force, the change in potential energy is a convenient way for evaluating the work done by the force. For example, if a mass m takes some complicated path (up, then down, then all around) in a uniform gravitational field g, the work done by gravity is simply -mg(h2 - h1) where h2 and h1 are the final and initial elevations, respectively. Similarly, for an electrostatic field E, voltage is the change in potential energy per unit charge. Physics texts provide the detailed equations: work as the integral of the dot product of force and differential distance, conservative force equals the negative of the gradient of a potential, no circulation for a conservative force, etc. Energy balances in engineering texts almost always consider the work from gravity as a difference in potential energy, and consider the work term for other forces such as pressure, sliding friction, etc.

Departing from your example of a ball thrown upwards, let's write the equation:

$$\frac{1}{2}mv^2 - \frac{GMm}{r} = cte$$ that states that the sum of potential and kinetic energy is constant.

Considering only velocity in radial direction, and making the derivatives with respect to time:

$$mv.\frac{dv}{dt} + \frac{GMm}{r^2}\frac{dr}{dt} = 0$$

But $${dr}{dt} = v$$ in this case, and cancel with the left side.

$$F = ma = m\frac{dv}{dt} = -\frac{GMm}{r^2}$$

And that is the Newton equation for gravity.

So the meaning for the sum of a potential term and the kinetic energy be a constant, is that its derivative results in the Newton's law. It is the same law in integral form.

It might help if you explain why you understand how kinetic energy can be converted to thermal energy so we can maybe better answer your question about the ball. I will assume you're thinking of, say, a block sliding on a surface with friction where the motion of the block causes it to rub against the surface thereby heating it. And I assume that with a ball thrown up in the air your question relates to the fact that the ball doesn't have a touching, direct interaction with the earth.

As you acknowledge, gravity is involved. As others have mentioned gravity is doing negative work on the ball. It's not a touching type of interaction, as gravity is acting at a distance, but it's nonetheless there. But the example of the block moving on a surface is similar in that friction arises from the interaction between the charged particles of the block and the surface near the point of contact. So in that sense it's similar to the ball/earth example as the force interacts at a distance.