Fundamental concepts of Scalar Kinematics

Scalar Kinematics is an important branch of Mechanics and is responsible for describing the motion of bodies without worrying about their causes.

The speed sign in the figure indicates that cars can travel a maximum of 80 km per hour on that stretch.
What is Scalar Kinematics?

Scalar kinematics is one of the main branches of mechanics . It is the area that studies the movement of bodies without attributing a cause to them. The word scalar refers to the fact that we deal with movements that are only one-dimensional, that is, movements that develop only along one direction of space , thus dispensing with the vectorial treatment of the physical quantities involved.

For the study of Scalar Kinematics, some concepts are of great importance, therefore, we will deal here with those that are fundamental for its understanding.

Fundamental concepts of Scalar Kinematics

→ Body : It is a limited portion of matter and is made up of particles, but it can be treated macroscopically as a single body within the scope of Scalar Kinematics.

→ Material point: It is every body whose dimensions can be neglected in relation to the distances involved. Some examples: The Earth moving around the Sun; a truck traveling between two distant cities; a ferry moving along a river, etc.

→ Extensive body: It is any body whose dimensions are comparable to the scales involved. In that case, they cannot be ignored. Some examples: The Earth in relation to the Moon; the movement of a truck leaving a garage; a person getting on a ferry, etc.

→ Reference : It is the adopted reference system. From it, distances , widths , depths , etc. are measured. The frame of reference is the position of space occupied by the observer . For example: The radius of the Earth is measured from its core, therefore, in this case, the center of the Earth is the adopted reference.

→ osition : It is the space occupied by a body and is determined by the distance measured in relation to some reference. It can be given in meters , kilometers , centimeters or any other units that are consistent with the scales involved in the observation.

→ Rest : Whenever a body maintains its constant position in relation to some referential, we will say that this body is at rest in relation to it. For example: On a moving bus, we are at rest with respect to the seats . It is important to emphasize that there is no absolute rest , because no body will be at rest in relation to all possible references.

→ ovement : When the position of a body changes in relation to a given frame of reference, we will say that this body is in motion in relation to that frame of reference. For example : In a moving bus, as we are at the same speed as the bus, we are moving with respect to the ground .

→ Trajectory : It is the succession of positions occupied by the body in relation to a given reference. For example : Footprints left in the sand; car tire tracks, etc.

→ Displacement : It is the difference between the initial and final position of a body in relation to some reference. In cyclic movements or in closed trajectories, the displacement will always be zero. For example : During a year, the displacement of the Earth in relation to the Sun is zero, because its trajectory is closed. It is defined by:

ΔS = S f – S 0

f: = End position;
0 = Starting position.

→ Distance traveled : It is the sum of the modules of all the distances traveled during a movement.

→ Average velocity : It is the ratio of the variation of the position by the time interval of a determined movement. It is defined by the equation:

Vm = ΔS         Δt _

ΔS = position variation (or displacement);
Δt = time interval.

→ Instantaneous speed : It is the speed indicated by the speedometer of the cars. It is an average speed, but for very small time intervals, close to zero. Its mathematical definition is similar to that of average speed:

          Δt

Δt ≈ 0
ΔS = position variation (or displacement);
Δt = time interval.

→ Uniform Movement (MU) : It is any movement in which the instantaneous velocity is always equal to the average velocity ; in this case , we say that the mobile travels equal distances in equal time intervals .

→ Time function of the MU position: It is a 1st degree mathematical equation used to describe the position ( S) of a mobile in relation to some reference (S 0 ) as a function of the elapsed time ( t). It is defined by the following equation:

S = S 0 + vt

S = End position;
0 = Initial position;

v = average speed;
t = instant of time.

→ Progressive movement: It is a movement in which the distance increases in relation to some reference, that is, when moving away. In this case, the velocity has a positive magnitude .

V > 0

→ Regressive or retrograde movement : It is a movement in which the distance of a mobile in relation to some reference decreases. The velocity in retrograde motion is negative .

V < 0

→ Average acceleration: It is the rate of change of velocity for a given time interval. The faster the velocity of a body changes as a function of time, the greater the magnitude of its acceleration.

If a body is increasing its speed, its acceleration will be positive and this motion will be called accelerated motion . If the speed of the mobile is decreasing, the movement is slowed down . The mathematical definition of acceleration is given by the following equation:

m = Δv
        Δt

Δv = speed variation;
Δt = time interval.

→ Instantaneous acceleration : It is the rate of change of velocity for very small time intervals , close to zero. Its mathematical definition is given by:

ins = Δv
          Δt

Δt ≈ 0
ΔS = speed variation;
Δt = time interval.

→ Uniformly Varied Motion (MUV) : It is a motion whose speed increases or decreases constantly . It is equivalent to saying that the acceleration does not change. It can be described by the following equations:

⇒ Hourly function of the MUV position : It is a 2nd degree mathematical equation that relates the position of the body in relation to some reference as a function of time. It is mathematically defined by:

S = S 0 + v 0 .t + At 2
                      2

0 = starting position;
0 = initial speed;

A = average acceleration;
t = instant of time.

⇒ Hourly function of velocity: It is a 1st degree mathematical equation that relates the velocity of a mobile in relation to a reference in a certain time interval . It is defined by the following equation:

v = v 0 + at

v = final speed;
0 = initial speed;

a = average acceleration;
t = instant of time.

⇒ Torricelli’s equation: It is a mathematical equation obtained from the combination of the two previous equations. It allows calculating variables such as final velocity initial velocity , acceleration and displacement of a mobile when we do not have information about the elapsed time intervals. It is therefore an extremely useful equation for the study of uniformly varied motion.

2 = v 2 + 2.A.ΔS

2 = final velocity squared;
2 = initial velocity squared;

A = average acceleration;
ΔS= displacement.

→ SI units. : In the International System of Units, position is given in meters (m), time interval is given in seconds (s), velocity is given in meters per second (m/s) and acceleration is given in meters per second squared ( m/s 2 ).

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