Kinematics
By: Valdivino Oliveira, 11-13-2022
Kinematics is the science of studying a body's motion, considering its position in space or distance traveled, velocity, and acceleration. Whether in straight or curvilinear motion.
Fundamental quantities
Distance (d)
It is the measurement, starting from an initial position to a final one and its unit according to the international system (IS) is the meter, represented by the letter m.
Figure 1 - A person walking along the path. The dashed line represents the distance.
Position or Displacement
It is the path in a straight line of a body, leaving an initial position to a final one in space. Its unit of measurement as well as distance is the meter. It is a vector quantity and must be expressed through its module (numerical value), direction, and sense.
Figure 2 - A person walking along the path. The dashed line represents the offset if it points in your direction and sense.
Velocity (v)
It is the measure of the variation of distance in the time of a body in motion. Its measurement unity is the meter per second (m/s) or kilometer per hour (km/h), according to the IS. Regarding its understanding, velocity can be scalar or vector, depending on the way it is interpreted. In several kinds of literature are found the terms, average speed, scalar average speed, instantaneous speed, etc.
Certainly, the purpose of creating so many nomenclatures would be to simplify understanding, but unfortunately, sometimes the definitions of each one impair it. Therefore, I will only deal with the said vector and scalar velocity here. The interpretation I will leave it to the reader to seek improvement in their studies.
Scalar velocity (speed)
A quantity is said to be scalar when to represent its value, we use only the dimension of its quantity, using a number, a scalar. The speed that is measured taking into account the ratio between the total distance traveled by a body on its trajectory and the time it took is called speed. This average does not take into account whether the body moves to the right or left, or whether the direction of motion is negative or positive, it just says “this body moves with an average speed of X (m/s)”. This quantity is nonnegative.
Vector velocity
(1)
This velocity, as well as the fundamental concept, measures the ratio between a distance (which in this case, being more specific, represents the displacement of the body) and the time spent, taking into account its module, direction, and sense. Its measurement becomes more precise as the displacement variation ∆s and time variation ∆t are minimal, approaching zero.
(2)
In cases where the variations of s and t tend to zero, we can say that this is the instantaneous velocity of the object.
Acceleration
It is a measure of the change in velocity over time. When this variation tends to zero, we have the acceleration at each instant of the trajectory. As it depends on a change in velocity, it is also a vector quantity, whose unit of measurement is m.s-1 according to the IS.
or using differential calculus,
(3)
If the acceleration of the body or system is constant, we can directly determine the equations that give us the relations of motion at each instant of time. Thus, by equation (2) we have,
(4)
Where s the final position and, the nitial position of a body. Opening the equation (3), we have,
(5)
If we want to have a relationship between acceleration and displacements, we look at the motion graph,
Figure 3 - Graph of velocity as a function of time.
, replacing (5) in this expression,
(6)
We now have a way of relating displacement, velocity, and acceleration, when the latter is constant. All the equations cited depend on the elapsed time, but if we join (4) and (6), isolating the time in each one, we find Torricelli's equation.
(7)
With this set of mathematical relationships, it is possible to analyze almost any type of situation involving motion with uniform acceleration.
Trabalhando com vetores
Tendo aprendido sobre as grandezas fundamentias no estudo da trajetória dos corpos, vamos amadurecer nosso conhecimento e representar as coisas como realmente são, pois, vivemos em um mundo tridimensional e precisamos incorporar a forma vetorial às equações de movimento estudadas.
Reescrevendo a equação (4), (5) e (6) teremos,
Figura 4 - Representa o deslocamento, no espaço tridimensional, partidno de um ponto B para um ponto A.
