Kinetic energy and gravitational potential energy

1. Kinetic energy and gravitational
potential energy
Let's define momentum
The momentum corresponds to a vector quantity that is defined by
[ overrightarrow { p } = m times overrightarrow { v } ]
which will depend on the study reference.
Moreover, by using additivity, one is able to define the momentum of a non-point body or
material system. As a result, it becomes possible to demonstrate that the quantity of matter
is equal to the quantity of movement of the center of inertia of the object studied affected by
the total mass of the system . So we have
[ overrightarrow { P } = M times overrightarrow { v _ { c } } ]
where C is the center of inertia. We use kg.ms -1 as a unit.
In a logical and natural way, the notion of quantity of movement is introduced into dynamics.
Indeed, the fundamental relation of the dynamics expresses the fact that the action of an
external force on a system leads to a variation of its momentum by the following expression:
One can also say of the notion of momentum that it is part, in the same way as energy, of
the quantities which are conserved in an isolated system and therefore a system which is not
subject to any external action or else these same external actions are negligible or
compensate for each other. This property is frequently used in collision theory.
In the case of an electromagnetic field, the momentum is called momentum. It then refers to
the volumetric pulse density of the field given by the formula:
Let's define a kinetic energy field
Kinetic energy is the energy related to the motion of a body . In fact, it is equal to the work
necessary to make the said body pass from rest to its movement of translation or rotation. It
therefore depends on both the speed of the object and its mass. Given that the speed of an

2. object depends on the reference frame chosen, it is also the case of the kinetic energy.
Kinetic energy is noted Ec and is expressed in joules (J).
In other words, any moving body has one . It can be macroscopic : it then depends on the
speed of the body in motion, and therefore on the reference frame of microscopic study : it is
linked to molecular agitation. An increase in microscopic kinetic energy results in an increase
in temperature.
In summary, kinetic energy is the energy of an object in motion.
Ec corresponding to the kinetic energy of the studied object with the unit Joule denoted J;
m corresponding to the mass of the object studied with the kilogram as unit noted kg;
And v corresponding to the speed of the object studied with a unit meter per second noted
ms -1 .
Example
Let us take the case of a system in translation. It is important to know that the relation
defining the kinetic energy does not apply for solids in rotation.
Example: The kinetic energy of a car that weighs 1 tonne and is traveling at 130 km/h is
Speed ​
​
is a physical quantity that is defined by an evolution over time.
The speed does not only define the speed of movement but can also correspond to the
speed of chemical reaction or even a speed of drying for example.
As a general rule, a speed is equal to the division of the measurement of a variation such as
a length, a volume or even a weight by the measurement of the time elapsed during this
variation.
The simplest example is movement speed. It is a distance divided by a time such as meters
per second or kilometers per hour.
The mass
In physics, mass corresponds to a positive and intrinsic physical quantity of a body.

3. More precisely, in Newtonian physics, mass corresponds to an extensive quantity. This then
means that the mass of a body formed of parts corresponds to the sum of the masses of
these different parts which compose it.
Moreover, it is essential to note that mass is a conservative quantity. As a result, it remains
constant in the case of an isolated system which therefore does not exchange matter with its
environment.
Let's define a solid in translation
It is a solid whose movement does not involve rotation : at a given instant each of its points
moves in the same direction, the same direction and at the same speed.
Expression of the kinetic energy of a solid in translation
If an object of mass m moves at a speed v following a translational motion then its kinetic
energy Ec is given by the formula:
Ec to joule (J)
m to kilogram (kg)
v in meters per second (ms -1 )
Be careful not to forget the square at speed!
Example
Consider an object of mass m = 6.7 g. This object is in translational motion at a speed v of
2.7 ms-1. What is the value of its kinetic energy? We know that v = 2.7 ms-1 and that m =
6.7g. First, convert the mass to kilograms. We therefore have: m = 6.7 g = 0.0067 kg. Thus,
we can apply the formula previously stated: Ec = 1/2 xmx v² Hence: Ec = 1/2 x 0.0067 x
(2.7)² Therefore: Ec = 0.024 J The kinetic energy of our object is therefore 0.024 Joules.
Let's define the gravitational potential
energy
It is the energy related to the weight of a body . It is due to the fact that this body is in a field
of gravity. The latter is exerted on any body having a mass and being near the Earth. It
therefore depends on the mass of the body and its altitude. It is noted Epp and is expressed
in joules like all other energies.
In other words, it depends on the relative position of the different parts of the system : only a
deformable system can possess, on a macroscopic scale, potential energy.

4. In summary, potential energy is the energy contained in an object above the ground.
Ep corresponding to the potential energy of the studied object with the unit Joule noted J;
m corresponding to the mass of the object studied with the kilogram as unit noted kg;
g corresponding to the intensity of gravity with the unit Newton per kilogram noted N.kg -1 .
Note that the approximate value of g is 9.81 N.kg -1 .
And v corresponding to the speed of the object studied with a unit meter per second noted
ms -1 .
The gravitational field can be
compared to an electric field
In classical physics, the gravitational field or gravitational field is a field distributed in space
and due to the presence of a mass capable of exerting a gravitational influence on any other
body present nearby (immediate or not). The introduction of this quantity makes it possible to
overcome the problem of the mediation of the action at a distance appearing in the
expression of the universal gravitational force.
One can interpret the gravitational field as being the modification of the spacetime metric .
The Newtonian approximation is then valid only in the case where the bodies have a low
speed compared to that of light in vacuum and if the gravitational potential that they create is
such that the quotient of the gravitational potential on the square of the speed of light in
vacuum is negligible.
We can approach the electric field and the gravitational field. Indeed, the expression of the
field and the potential are only different from a constant. Moreover, the main theorems of
calculations, that of superposition or of Gauss for example, can be applied in both cases.
What differentiates them then is the attractive character, therefore between two charges of
opposite sign, or repulsive, therefore between two charges of the same sign, of the electric
field, whereas the gravitational field can only be attractive.
The gravitational interaction force , like the electrostatic interaction force, is a conservative
force. Thus, they both represent the gradient of a potential energy. In this case, it is then

5. possible to adapt absolutely all the field and potential calculations studied within the
framework of the course on the distribution of masses in order to calculate the gravitational
field and potential at a defined point in space. . The same is true with Gauss's theorem.
It may be worth mentioning that the fundamental electric force , also called the Coulomb
force, can be used as the foundation of electrostatics. Thus, one can deduce from this
foundation Gauss's theorem.
This is therefore why we can say that the formal resemblance, that is to say the similarity of
the mathematical formulas, between the Coulomb force and the gravitational force is a solid
basis for founding the analogy between the two classes of phenomena described in this
course.
Thus, from the Coulomb force and by superposition, one may be able to establish integral
expressions of the electric field as a function of the charge distribution. Although these
calculations are too complex to be useful in analytical calculations, they can be very useful in
order to determine an electric field by numerical resolution, that is to say by computer.
Note that it is possible to prove these formulas using the superposition theorem.
In physics, we call an electric field any vector field created by electrically charged particles.
More exactly, when we are in the presence of a charged particle, the local properties of the
defined space are then modified which makes it possible to define the notion of field. Indeed,
if another charge happens to be in the said field, it will undergo what is called the action of
the electric force which is exerted by the particle despite the distance. The electric field is
then said to be the mediator of said action at a distance.
If we want to be more precise, we can define in a defined Galilean frame of reference, a load
q defined with a velocity vector v which undergoes from the other loads present, whether
fixed or mobile, a force that we will define as force by Lorentz. This force breaks down as
follows:
the magnetic field. This thus describes the part of the force exerted on the load which
depends on the displacement of this same load in the chosen frame of reference.
Moreover, it is important to note that the two fields, electric and magnetic, depend on the
reference frame of study.
With this formula, we can then define the electric field as being the field translating the
remote action undergone by a fixed electric charge in a reference frame defined by all the
other charges, whether mobile or fixed.
But we can also define the electric field as being any region of space in which a charge is
subjected to a so-called Coulomb force.
We begin to speak of an electrostatic field when, in a reference frame of study, the charges
are fixed. Note also that the electrostatic field does not correspond to the electric field as
described earlier in this article since indeed, when the charges are moving in a frame of

6. reference, it is necessary to add to this frame
of reference an electric field which is induced by the displacement of the charges in order to
obtain a complete electric field.
But, the electric field remains in reality a relative character since it cannot exist
independently of the magnetic field. Indeed, if we observe the correct description of an
electromagnetic field, it involves a four-dimensional electromagnetic field tensor whose
temporal components then correspond to that of an electric field. Only this tensor has a
physical meaning. So, in the case of a change of reference frame, it is quite possible to
transform a magnetic field into an electric field and vice versa.
One speaks of electrostatic fields when the charges which constitute the field are at rest in
the reference frame of study. This field is therefore deduced from the expression of
Coulomb's law, also called electrostatic interaction.
In physics, the electromagnetic field is the representation in space of an electromagnetic
force exerted by charged particles. This field then represents all the components of the
electromagnetic force which apply to a charged particle which then moves in a Galilean
frame of reference.
We can then define the force undergone by a particle with charge q and velocity vector by
the following expression:
the magnetic field. This thus describes the part of the force exerted on the load which
depends on the displacement of this same load in the chosen frame of reference.
Indeed the separation of the magnetic part and the electrical part depends only on the point
of view taken according to the reference frame of study.
Moreover, it can be interesting to know that the equations of Maxwell govern the two coupled
components, i.e. electric and magnetic, so that any variation of a component will induce the
variation of the other component.
Moreover, the behavior of electromagnetic fields is classically described by Maxwell's
equations and more generally by quantum electrodynamics.
The most used way to define the electromagnetic field is that of the electromagnetic tensor
of special relativity.
Expression of the potential energy of gravity

7. If a body of mass m is located at an altitude z then its gravitational potential energy Epp can
be calculated using the following formula:
Epp is in joules (J)
Epp is in joules (J)
*g the intensity of gravity is in N.kg -1
**z1 and z0 is in meters (m)
*The intensity of gravity is equal to approximately 9.81 m s −2 (or 9.81 N/kg). ** z1 and z0
being respectively the points of arrival and departure altitude of the object Note: The altitude
is expressed compared to a reference which must be chosen before calculating the potential
energy. However, the most frequent landmark is that associated with the ground. In an
exercise statement, normally, this is specified.
Example
Let's say we throw a soccer ball that gets stuck in a tree. This balloon has a mass of 400 g. It
remains stuck at a height of 3.7 m. What is its gravitational potential energy? We know that
m = 400 g and z1 = 3.7 m. We first need to convert the mass to kilograms. We therefore
have m = 400 g = 0.4 kg. We can therefore apply the relationship previously stated: Δ (Epp)
= mxgx (z1 - z0) Hence: Δ (Epp) = 0.4 x 9.81 x 3.7 (here z0 = 0 since the ball left of the
ground) So: Δ(Epp) = 14.52 J. Thus, the gravitational potential energy of our balloon is 14.52
Joules! Note: The altitude is expressed in relation to a reference which must be chosen
before calculating the potential energy. However, the most frequent landmark is that
associated with the ground.
Let's define the inertia of a body
In physics, one calls inertia of a body , in a Galilean frame of reference, a tendency of this
body to preserve its speed . Indeed, when there is absence of external influences, we also
speak of external forces, then any body that we consider as punctual will continue in a
uniform rectilinear movement.
Note that we also call inertia, principle of inertia or law of inertia. Then, when Newton arrived,
it was also called Newton's first law.
We call Galilean referential any referential within which the principle of inertia is verified

8. Even if there is no Galilean frame of reference in the strict sense. It is however possible to
consider certain usual reference frames as Galilean if certain conditions are verified:
Thus, the terrestrial frame of reference can be considered Galilean if we consider a
movement whose duration does not exceed a few minutes in order to overcome the Earth's
own rotational movement.
The geocentric frame of reference can also be considered as being Galilean if we consider a
movement whose duration does not exceed a few hours in order to overcome the movement
of rotation of the Earth around the Sun.
The heliocentric reference frame can also be considered as being Galilean because the
impact of the rotational movement of the Sun within the galaxy is negligible.
.