The paradox of gravity not respecting the energy conservation principle

Bernoulli’s theorem

A body orbiting around a planet is in equilibrium between two forces: the centrifugal force and the gravitational force

where Fc is the centrifugal force, Fg is the gravitational force, m1 is the mass of the orbiting body, m2 is the mass of the planet, v is the speed of the body while orbiting, r is the distance of the body from the center of the planet, G is the gravitational constant

The orbiting body is characterized by a potential energy called “gravitational” caused by the field in which it is submerged. When speaking about a potential energy, mind immediately runs to Bernoulli’s theorem that states that, for a liquid, the sum of potential, kinetic and pressure energy is constant. You could think to the water contained in a basin that is situated on the top of a mountain, water that is forced to pass into a conduct, transforming, due to the altitude, the initial potential in kinetic and pressure energy, later collected into a turbine in order to transform energy into electricity.

Bernoulli’s theorem is an application of the principle of conservation of energy . The energy changes  in form but its total amount  doesn’t change. In the case of a forced conduct the potential energy of the water in altitude U=mgh is transformed in kinetic energy Ec=1/2mv^2 and in pressure energy.

Energy conservation principle and the paradox with gravity

The principle of energy conservation could be in the same way applied to a body orbiting in the gravitational field of a planet. In a gravitational field the potential energy is expressed by the general formula .

The expression U=mgh (to which we all were used at school) is a particular case of the foregoing more general expression and can be applied only in the case of h<<R where R is the Earth radius.

The energy conservation principle for a body in a gravitational field is expressed by the relation:

The total amount of energy is the sum of kinetic and potential energy.

Thus, according to this relation, a body in free fall in the gravitational field of a planet, will convert its potential energy  into kinetic energy, but maintaining constant the sum of the two and producing an increase in speed.

The total amount of energy will remain the same. The opposite  could not be possible : take a body,  that, with a certain starting potential  (but without possessing any kinetic energy)  increases its potential even without receiving any external addition of energy. This result will be obtained  only by diminishing the kinetic factor: in fact , in order to increase the potential energy, the kinetic  has to decrease, but when this energy is already zero…it can’t become negative. Thus, in a hydroelectric power plant, water is driven  nightlong up on altitude to the eventual lake  by spending electric energy that, during the night, has a lower cost. However, the necessity is to spend  farther energy in order to obtain  water again and in a greater quantity of potential energy to be used daylong  to produce electrical current (to be sold at a higher price) . Another example could be relating to a chute on which you can slide downwards without any effort  but when going up the opposite direction you have to add a good amount of energy to respect the  conservation law.

Let’s imagine  a similar situation with respect to an orbiting body moving  in an orbital direction only and not  in any whatsoever radial direction. The body, thus, possesses potential energy only, being its speed perpendicular to the radial direction, on which the gravitational force is acting upon, and , incidentally, considering all factors, this datum cannot have any influence on  calculation.

Consider now a meteorite that happens to hit the orbiting body in a direction tangential to the orbit, going in this manner, to increase the speed of the satellite we are taking into consideration (but let us suppose  with a very small increase). Wishing to make a comparison with the forced conduct, imagine you were trying to launch a little amount of water from a bucket upward in the air just by impressing to the water a small  kinetic energy (that won’t, anyway, be sufficient to win the gravitational force). Similarly  water rises a little through the conduct but then  it necessarily falls down again, because there is not force enough to pull  water  to the altitude of the lake. In the same way  the new speed acquired by the considered satellite  will be V’=V+ΔV, where ΔV is very small. Since, however, V’>V, the centrifugal force grows a little according to the relation .

The gravitational force, on the other hand,  will remain the same. The equilibrium will be lost, the satellite  acquiring a force F resulting= Fc’-Fg  sufficient to drag  it away from the planet.

The resulting force will originate a speed in the radial direction, in such a manner that the kinetic energy, moving in the radial direction, would start increasing and the body would start departing from the planet.

Since the ditance grows, as far as the body is departing from the planet, Fc’ decreases in proportion to 1/r. In the same time the gravity force will decrease faster and faster, in proportion to 1/r^2. The body would  accelerate more and more and the kinetic energy would grow very fast , no energy  by the exterior being added (or very low energy).  At the same time, since the body would be departing from the planet,  the potential energy  should grow…In the same manner the kinetic energy should increase, and so the total energy.

And here the paradox starts. The body should immediately stop orbiting around the planet and be trapped in another orbit, just because, departing from the original planet, the potential energy would increase, making the kinetic  decrease according to the conservation energy principle. But, however, the centrifugal force, continuing to be higher than the gravitational force,  the body should keep departing with a spiral movement from the planet.

Now, I’m not a physicist. I don’t pretend to be right. But someone has to explain me this paradox. Could you help me?

Bye.