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Understanding Physical Quantities: Scalars and Vectors

 

Understanding Physical Quantities: Scalars and Vectors

 

Subject: Physics

Prepared By: Ma. Johanna B. Testa

 

Topic: Scalar and Vector Quantities (Linear Addition)

 

I. Learning Objectives

By the end of this lesson, students will be able to:

  • Define physical quantities and differentiate between scalar and vector quantities.
  • Identify examples of scalar and vector quantities in daily life and scientific contexts.
  • Apply rules to add scalar quantities.
  • Apply graphical and algebraic methods to add linear vector quantities.

 

II. Lesson Content

1. Physical Quantities

Definition:

A physical quantity is a property of a material or system that can be quantified by measurement. It consists of a numerical value and a unit. Examples include length (meters), mass (kilograms), time (seconds), and force (Newtons).

 

Classification:

Physical quantities are broadly classified into:

Scalar Quantities

Vector Quantities

 

2. Scalar Quantities

Definition:

A scalar quantity has only magnitude (size) and no direction. Scalars are described completely by a numerical value and a unit.

 

Examples:

Distance: The total path length traveled, irrespective of direction.

Speed: The rate of change of distance with time.

Mass: The amount of matter in an object.

Temperature: A measure of the thermal energy of a system.

Time: The ongoing sequence of events taking place.

Source: Khan Academy

 

Addition of Scalars:

Scalar quantities are added algebraically, considering only their magnitudes.

 

Example:

A runner completes two laps of 400 meters each.

Total distance = 400 m + 400 m = 800 m

 

3. Vector Quantities

Definition:

A vector quantity has both magnitude and direction. Vectors are represented graphically by arrows; the length denotes magnitude, and the arrowhead indicates direction.

 

Examples:

Displacement: The change in position of an object.

Velocity: The rate of change of displacement with time.

Acceleration: The rate of change of velocity with time.

Force: An interaction that changes the motion of an object.

Momentum: The product of an object's mass and velocity.

Source: Britannica

 

Representation:

Vectors are often denoted by boldface letters (e.g., v) or with an arrow above the letter (e.g., 𝑣).

 

4. Addition of Vector Quantities (Linear/One-Dimensional Only)

In linear (one-dimensional) motion, vector addition simplifies to algebraic addition, taking direction into account.

Case 1: Vectors in the Same Direction

Add magnitudes directly.

Direction remains the same.

Example:

A car moves 30 m east, then 20 m east.

Resultant displacement = 30 m + 20 m = 50 m east

Case 2: Vectors in Opposite Directions

Subtract the smaller magnitude from the larger one.

Direction is that of the larger vector.

Example:

A person walks 70 m east, then 40 m west.

Resultant displacement = 70 m - 40 m = 30 m east

Source: Khan Academy

 

III. Sample Problems

A. Scalar Quantities

A bus travels 25 km in the morning and 35 km in the afternoon. What is the total distance?

Solution:

25 km + 35 km = 60 km

 

B. Vector Quantities (Linear Only)

A girl walks 40 m north, then 20 m north again. What is her total displacement?

Solution:

40 m + 20 m = 60 m north

 

A cyclist goes 100 m east, then 60 m west. What is the resultant displacement?

Solution:

100 m - 60 m = 40 m east

 

IV. Assessment Questions

  1. Define scalar and vector quantities.
  2. Provide two examples each of scalar and vector quantities.
  3. If a man walks 100 m north, then 100 m south, what is his total distance and displacement?
  4. Two forces, 30 N east and 20 N west, act on an object. What is the net force?

 

V. Summary

  • Scalars have only magnitude and are added algebraically.
  • Vectors have both magnitude and direction; in linear motion, they are added considering direction.
  • Understanding the distinction between scalars and vectors is crucial in physics to analyze motion and forces accurately.

 

VI. References

  • Britannica. (n.d.). Vector analysis. Retrieved from https://www.britannica.com/science/vector-analysis
  • Khan Academy. (n.d.). Scalars and vectors. Retrieved from https://www.khanacademy.org/science/ap-college-physics-1/xf557a762645cccc5%3Akinematics/xf557a762645cccc5%3Ascalars-and-vectors-in-1d/a/scalars-and-vectors
  • Khan Academy. (n.d.). Adding and subtracting vectors. Retrieved from https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf%3Avectors/x9e81a4f98389efdf%3Avector-add-sub/v/adding-and-subtracting-vectors