Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Position, distance and displacement.
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Position, distance and displacement.
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Subject: Physics
Class: Senior Secondary 2
Term: 1st Term
Week: 1
Theme: Interaction Of Matter, Space And Time
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Students shouldbe able to use the Cartesian systemto locate the position of objecton the x-y plane.
If the student started at the hostel and ended up in the classroom, the displacement would be the straight line from hostel to classroom, regardless of the path taken.
Worked Examples (Nigerian Context): Example 1: Locating a Market Stall Imagine a new open-air market in Kaduna is designed on a grid system, with the main entrance as the origin (0,0). A food vendor's stall, Mama Bintu's kitchen, is located 4 units east and 3 units north of the main entrance.
Question: What are the coordinates of Mama Bintu's kitchen?
Solution: East corresponds to the positive x-direction. North corresponds to the positive y-direction. So, x = 4, y =
3. The coordinates are (4, 3).
Example 2: Locating a Farm Section A large cassava farm in Benue State uses a coordinate system for managing its sections. The main farm office is at the origin (0,0). A section of the farm prone to erosion is located 2 km west and 5 km south of the main office.
Question: What are the coordinates of the erosion-prone section?
Solution: West corresponds to the negative x-direction. South corresponds to the negative y-direction. So, x = -2, y = -
5. The coordinates are (-2, -5).
Example 3: Identifying a Landmark On a tourist map of Abuja, a famous monument, the Millennium Tower, is at coordinates (-3, 4) if the Aso Rock Presidential Villa is taken as the origin (0,0).
Question: Describe the position of the Millennium Tower relative to Aso Rock.
Solution: The x-coordinate is -3, meaning 3 units in the negative x-direction (west). The y-coordinate is 4, meaning 4 units in the positive y-direction (north). * Therefore, the Millennium Tower is located 3 units west and 4 units north of Aso Rock. --- This section provides a detailed explanation of position, with a focus on its representation using the Cartesian coordinate system. Distance and displacement are introduced briefly as related but distinct concepts to be covered in subsequent lessons. 2.
1. Position Position refers to the location of an object relative to a specific reference point. It answers the question, "Where is the object?" To define an object's position unambiguously, a frame of reference is essential.
Reference Point/Origin: This is a fixed, chosen location from which all other positions are measured. It acts as the starting point (the "zero" point). Without a reference point, "position" is meaningless. For example, telling someone your house is "far" is vague; saying it's "200 meters north of the community borehole" makes the borehole the reference point. 2.
2. The Cartesian Coordinate System The most common and effective way to define position in a two-dimensional plane is using the Cartesian coordinate system, also known as the rectangular coordinate system.
Components: X-axis (Horizontal Axis): This is a horizontal number line. Positive values are typically to the right of the origin, and negative values are to the left.
Y-axis (Vertical Axis): This is a vertical number line. Positive values are typically above the origin, and negative values are below. Origin (0,0): The point where the X-axis and Y-axis intersect. This is the designated reference point from which all positions are measured. Ordered Pair (x, y): The position of any point on the Cartesian plane is represented by an ordered pair (x, y), where 'x' is the horizontal coordinate (abscissa) and 'y' is the vertical coordinate (ordinate). The 'x' coordinate indicates the position along the X-axis relative to the origin. The 'y' coordinate indicates the position along the Y-axis relative to the origin.
The order matters: (2, 3) is different from (3, 2).
Quadrants: The X and Y axes divide the plane into four regions called quadrants.
Quadrant I: x > 0, y > 0 (e.g., (3, 2))
Quadrant II: x 0 (e.g., (-3, 2))
Quadrant III: x 0, y < 0 (e.g., (3, -2)) 2.
3. How to Plot a Point (x, y) on the Cartesian Plane: To plot a point (x, y):
1. Start at the Origin (0,0).
2. Move horizontally: Move 'x' units along the X-axis. If 'x' is positive, move to the right. If 'x' is negative, move to the left.
3. Move vertically: From that new position (on the X-axis), move 'y' units parallel to the Y-axis. If 'y' is positive, move upwards. If 'y' is negative, move downwards.
4. Mark the point: This final location is the point (x, y). 2.
4. Position as a Vector Quantity Position is a vector quantity because it has both magnitude (how far from the origin) and direction (in which direction from the origin). For example, a position of (5, 3) means 5 units in the positive x-direction and 3 units in the positive y-direction from the origin. 2.
5. Brief Distinction: Distance and Displacement (To be covered in detail in subsequent lessons)
Distance: Distance is a scalar quantity that refers to the total path length covered by an object during its motion. It only has magnitude. For instance, if a student walks 50m from their hostel to the dining hall and then 30m to the classroom, the total distance covered is 80m.
Displacement: Displacement is a vector quantity that refers to the shortest straight-line distance between an object's initial position and its final position, along with the direction. It has both magnitude and direction. If the student started at the hostel and ended up in the classroom, the displacement would be the straight line from hostel to classroom, regardless of the path taken.
Worked Examples (Nigerian Context): Example 1: Locating a Market Stall Imagine a new open-air market in Kaduna is designed on a grid system, with the main entrance as the origin (0,0). A food vendor's stall, Mama Bintu's kitchen, is located 4 units east and 3 units north of the main entrance.
Question: What are the coordinates of Mama Bintu's kitchen? * Solution: 3.
1. Teacher Activities: Introduction (5 minutes): Initiate a discussion on how people describe locations in everyday life (e.g., "turn left at the roundabout," "opposite the church"). Introduce the need for a more precise, universal method to describe location, especially in physics. Briefly mention the theme "Interaction of Matter, Space and Time" and how position is fundamental to understanding movement in space. Concept Explanation and Demonstration (15 minutes): Draw a large Cartesian plane on the whiteboard or project one using a projector. Clearly label the X-axis, Y-axis, and Origin (0,0). Explain the concept of an ordered pair (x, y) and how each coordinate relates to horizontal and vertical movement. Demonstrate step-by-step how to plot several points in different quadrants (e.g., (2,3), (-4,1), (-1,-3), (5,-2)). Emphasize starting at the origin and moving horizontally first, then vertically. Demonstrate how to identify the coordinates of points already plotted on the graph. Briefly introduce distance and displacement as concepts distinct from position, highlighting that position is the current location, while distance/displacement relate to change in position.
Interactive Guided Practice (15 minutes): Call on individual students to come to the board and plot points or identify coordinates. Use the "think-pair-share" strategy: Present a coordinate, ask students to plot it on their individual graphs, discuss with a partner, then share with the class.
Ask questions like: "What does a negative x-value mean?" or "Where would a point (0,5) be located?" Activity Facilitation (10 minutes): Divide students into small groups for a practical plotting activity (see student activities). Circulate around the classroom, providing support, checking for understanding, and correcting misconceptions.
Consolidation and Wrap-up (5 minutes): Review the main concepts of position and the Cartesian system. Reinforce the importance of a reference point and the ordered pair (x, y). Briefly connect the day's lesson to the upcoming topics of distance and displacement. 3.
2. Student Activities: Discussion and Brainstorming: Students participate in a class discussion about methods of describing location.
Note-Taking: Students take notes on the definitions of position, reference point, and the components of the Cartesian system.
Drawing and Labeling: Students draw their own Cartesian planes in their notebooks, labeling axes, origin, and quadrants.
Individual Plotting Practice: Students plot various coordinates given by the teacher on their individual graphs as the teacher demonstrates.
Group Activity: "Mapping Our School Compound" (Practical Focus): Materials: Large graph paper or plain paper, rulers, pencils.
Instructions: In groups, students will imagine a simplified map of their school compound. They will choose a central point (e.g., the school gate, the principal's office) as their origin (0,0). They will identify and assign approximate coordinates to various key locations within the school compound (e.g., library, classroom block, science lab, football field, staff room, borehole). They will then plot these locations on a Cartesian plane they draw, ensuring their axes are clearly labeled (e.g., in "steps" or "meters"). Each group will present their "map" and explain how they determined the coordinates for at least three locations.
Peer Assessment: Students exchange their plotted graphs with a partner to check for accuracy in plotting points and labeling. --- The following questions are designed to reinforce the understanding of the Cartesian system and plotting points, directly addressing the performance objectives.
Question 1: Plot the following points on a Cartesian plane: A = (2, 5) B = (-3, 4) C = (0, -2) D = (4, -1) E = (-5, -3)
Solution 1: Steps: Draw an X-axis and a Y-axis intersecting at the origin (0,0). Label the axes and mark appropriate scales. For A (2, 5): Start at (0,0), move 2 units right, then 5 units up. Mark the point. For B (-3, 4): Start at (0,0), move 3 units left, then 4 units up. Mark the point. For C (0, -2): Start at (0,0), do not move horizontally (x=0), then move 2 units down. Mark the point. For D (4, -1): Start at (0,0), move 4 units right, then 1 unit down. Mark the point. For E (-5, -3): Start at (0,0), move 5 units left, then 3 units down. Mark the point.
Commentary: This exercise ensures students can apply the basic rule of plotting (x, y) coordinates across all four quadrants and on the axes.
Question 2: Consider a grid representing a section of the city of Ibadan, with the iconic Cocoa House as the origin (0,0). A Keke Napep driver picks up a passenger at position P and drops them off at position Q. If the passenger was picked up at 3 units East and 2 units North of Cocoa House, and dropped off at 1 unit West and 4 units South of Cocoa House, what are the coordinates of P and Q?
Solution 2: Steps: Identify the reference point: Cocoa House (0,0). East corresponds to positive x; North to positive y. West corresponds to negative x; South to negative y.
For P: 3 units East (x=+3), 2 units North (y=+2). So, P = (3, 2).
For Q: 1 unit West (x=-1), 4 units South (y=-4). So, Q = (-1, -4).
Commentary: This question integrates a real-world scenario (Nigerian transport) with the concept of assigning coordinates based on directional descriptions relative to an origin.
Question 3: On a Cartesian plane, a point F is located such that it is 6 units away from the Y-axis to the left, and 3 units away from the X-axis upwards. a. Write down the coordinates of point F. b. In which quadrant is point F located?
Solution 3: Steps: a. "6 units away from the Y-axis to the left" means x = -6. "3 units away from the X-axis upwards" means y = +
3. Therefore, the coordinates of F are (-6, 3). b. Since the x-coordinate is negative and the y-coordinate is positive, point F is located in Quadrant I
I. Commentary: This question tests the understanding of how distance from axes translates to coordinate values and reinforces knowledge of quadrants.
Question 4: A Nigerian oil company is mapping new exploration sites in the Niger Delta. They use a central drilling platform as their origin (0,0). They have identified three potential sites: Site 1: 5 km East, 2 km North.
Site 2: 3 km West, 4 km North.
Site 3: 1 km East, 6 km South. Plot these three sites on a Cartesian plane, assuming each unit represents 1 km.
Solution 4: Steps: Draw a Cartesian plane and label axes, ensuring a scale (1 unit = 1 km). Site 1 (5 km East, 2 km North) corresponds to coordinates (5, 2). Plot this point. Site 2 (3 km West, 4 km North) corresponds to coordinates (-3, 4). Plot this point. Site 3 (1 km East, 6 km South) corresponds to coordinates (1, -6). Plot this point.
Commentary: This example applies the concept of position to an industry-relevant scenario in Nigeria, requiring students to convert directional descriptions into numerical coordinates and then plot them. ---
GPS and Ride-Hailing Services (e.g., Uber, Bolt, Gokada): In Nigeria, these services rely heavily on GPS technology, which uses coordinate systems (specifically latitude and longitude, a spherical coordinate system) to precisely locate vehicles and passengers. A student can understand that when they request a ride, their phone sends its coordinates, and the driver's app uses coordinates to navigate to them. This helps in efficient transportation, reducing wait times and improving safety in bustling cities like Lagos, Abuja, and Port Harcourt. Urban Planning and Infrastructure Development: Nigerian urban planners and engineers use coordinate systems to map out new residential estates, industrial zones, and infrastructure projects like roads, bridges, and drainage systems. Before a new market is built in Kano or a new road is constructed in Enugu, precise coordinates are used to define boundaries, plan layouts, and ensure that buildings and utilities are placed in their designated locations, preventing disputes and ensuring optimal use of space.
Oil and Gas Exploration and Management: Nigeria's economy is heavily dependent on oil and gas. Companies involved in exploration and production use sophisticated GPS and coordinate systems to pinpoint the exact locations of oil wells, pipelines, and offshore platforms in the Niger Delta and beyond. This precision is critical for efficient drilling operations, monitoring environmental impacts, and managing resource extraction safely and effectively. For instance, the exact coordinates of a new oil exploration block are defined to avoid encroaching on existing facilities or protected areas. ---