# 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.

**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.

→ **P 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.

→ **M 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}**

**S _{f:} = End position;
S _{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;
S _{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:

**A _{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:

**A _{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**

**S _{0} = starting position;
v _{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;
v _{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.

**v ^{2} = v _{0 }^{2} + 2.A.ΔS**

**v ^{2} = final velocity squared;
v _{0 }^{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} ).