Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Projectile
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Projectile
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Subject: Physics
Class: Senior Secondary 2
Term: 1st Term
Week: 4
Theme: Interaction Of Matter, Space And Time
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Students shouldbe able to: identify aprojectilemotion Derive the range,maximumheight and time of flight
This section provides a detailed explanation of projectile motion, its components, and the derivation of key quantities. 2.
1. Definition of Projectile Motion Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The path followed by a projectile is called its trajectory.
Key Assumptions: For ideal projectile motion, air resistance is considered negligible, and the acceleration due to gravity (g) is assumed to be constant throughout the motion. The rotation of the Earth is also ignored. 2.
2. Examples of Projectile Motion A stone thrown horizontally from a cliff. A football kicked from the ground. A javelin or shot put thrown by an athlete. Water flowing out of a hosepipe. A bullet fired from a gun (ignoring air resistance over short distances). 2.
3. Components of Projectile Motion Projectile motion can be analyzed by resolving the motion into two independent components: Horizontal Component: The velocity in the horizontal direction (x-axis) remains constant throughout the motion because there is no horizontal force (assuming negligible air resistance). The acceleration in the horizontal direction (ax) is zero.
Formula: `x = uxt` where `ux` is the initial horizontal velocity and `t` is time.
Vertical Component: The motion in the vertical direction (y-axis) is affected by the acceleration due to gravity (g). The acceleration in the vertical direction (aγ) is `-g` (taking upward as positive). The velocity changes continuously due to gravity. At the maximum height, the vertical component of velocity (`vγ`) is zero. Relevant equations of motion (SUVAT equations) apply: `v = u + at` `s = ut + 1⁄2at2` `v2 = u2 + 2as` 2.
4. Initial Velocity and its Components If a projectile is launched with an initial velocity `u` at an angle `θ` to the horizontal: Horizontal component of initial velocity (ux): `ux = u cosθ` Vertical component of initial velocity (uγ): `uγ = u sinθ` 2.
5. Derivations of Key Quantities Let the initial velocity be `u` at an angle `θ` to the horizontal. Take upward direction as positive. So, acceleration `a = -g`. This section outlines practical activities for both teachers and students to facilitate understanding and engagement. 3.
1. Teacher Activities Introduction (10 minutes): Begin by asking students to brainstorm everyday examples of objects that are thrown or kicked and then fall back to the ground. (e.g., football, stone, basketball shot).
Perform a simple demonstration: Throw a duster or a small ball across the classroom, drawing attention to its curved path. Introduce the term "projectile motion" and explain that this curved path is called a "trajectory." Briefly state the assumptions made (negligible air resistance, constant g). Concept Explanation and Derivations (30 minutes): Explain the independence of horizontal and vertical components of motion. Use diagrams to resolve initial velocity `u` into `ux` and `uγ`. Guide students through the step-by-step derivation of the Time of Flight (T). Write the equations clearly on the board, explaining each substitution and simplification. Encourage students to participate by suggesting the next step or recalling relevant equations of motion. Similarly, guide the derivation of Maximum Height (H), emphasizing that `vγ = 0` at this point. Finally, guide the derivation of Range (R), linking it directly to the horizontal velocity and the derived Time of Flight. Discuss the significance of the angle of projection, particularly the 45° angle for maximum range.
Worked Examples (20 minutes): Present one or two worked examples (see Guided Practice Section below) on the board. Work through the problems step-by-step, explaining the choice of formula, substitution, and calculation. Encourage students to attempt parts of the calculation or identify the given and required quantities.
Recap and Q&A (5 minutes): Summarize the key formulae derived. Allow students to ask questions for clarification. 3.
2. Student Activities Brainstorming and Observation (10 minutes): Students actively participate in brainstorming examples of projectile motion. Students observe the teacher's demonstration and describe the path taken by the projectile.
Participation in Derivations (30 minutes): Students follow along as the teacher derives the formulae for T, H, and R, copying them into their notebooks. Students answer questions posed by the teacher during derivations, recalling equations of motion or suggesting algebraic steps. Students may be asked to complete a step in the derivation independently or in pairs.
Problem Solving (20 minutes): Students actively engage in solving the worked examples presented by the teacher, either by attempting them independently first or by following the teacher's steps. Students may be asked to solve similar problems in small groups or individually as classwork.
Discussion and Questioning (5 minutes): Students ask questions about concepts they find challenging. Students engage in short discussions about the real-life implications of projectile motion. --- The teacher should guide students through these problems, encouraging participation and explanation of steps. (Use `g = 10 m/s2` for simplicity unless specified).
Question 1 (Conceptual): A Nigerian student kicks a football across a field. a) Describe the type of motion the football undergoes once it leaves the ground until it hits the ground again. b) What are the main forces acting on the ball during its flight (ignoring air resistance)? c) How does the horizontal component of the ball's velocity change during its flight?
Solution 1: a) Projectile motion. The football is moving under the sole influence of gravity after being kicked. b) The only main force acting on the ball is gravity (its weight), acting downwards. c) The horizontal component of the ball's velocity remains constant throughout its flight because there is no horizontal force acting on it (ignoring air resistance). Question 2 (Application of formulae - Time of Flight & Max Height): A stone is thrown from the ground with an initial velocity of `20 m/s` at an angle of `30°` to the horizontal. Take `g = 10 m/s2`. a) Calculate the time taken to reach its maximum height. b) Calculate the maximum height reached by the stone.
Solution 2: Given: `u = 20 m/s`, `θ = 30°`, `g = 10 m/s2`.
Step 1: Resolve initial velocity into components. Vertical component (`uγ`) = `u sinθ` = `20 sin30°` = `20 0.5` = `10 m/s`. Horizontal component (`ux`) = `u cosθ` = `20 cos30°` = `20 0.866` = `17.32 m/s`. a) Time to reach maximum height (t_max): At maximum height, the vertical velocity (`vγ`) is `0 m/s`. Using `vγ = uγ + aγt_max`: `0 = 10 + (-10) * t_max` `0 = 10 - 10t_max` `10t_max = 10` `t_max = 1 second`.
Comment: This is half the total time of flight. b)
Maximum height (H): Using `vγ2 = uγ2 + 2aγH`: `02 = 102 + 2(-10)H` `0 = 100 - 20H` `20H = 100` `H = 5 meters`.
Comment: Alternatively, students can use `H = (u2 sin2θ) / (2g)` directly: `H = (202 (sin30°)2) / (2 10)` `H = (400 * (0.5)2) / 20` `H = (400 * 0.25) / 20` `H = 100 / 20 = 5 meters`.* Question 3 (Application of formulae - Time of Flight & Range): A javelin is thrown by an athlete in a school sports competition with an initial velocity of `30 m/s` at an angle of `60°` to the horizontal. Assume `g = 10 m/s2`. a) Calculate the total time the javelin spends in the air (time of flight). b) Calculate the horizontal distance covered by the javelin (range).
Solution 3: Given: `u = 30 m/s`, `θ = 60°`, `g = 10 m/s2`. a)
Time of Flight (T): Using `T = (2u sinθ) / g`: `T = (2 30 sin60°) / 10` `T = (60 * 0.866) / 10` `T = 51.96 / 10` `T = 5.196 seconds` (approximately `5.20 seconds`). b)
Range (R): Using `R = (u2 sin(2θ)) / g`: `R = (302 sin(2 60°)) / 10` `R = (900 * sin120°) / 10` `R = (900 * 0.866) / 10` `R = 779.4 / 10` `R = 77.94 meters` (approximately `77.9 meters`).
Comment: Alternatively, using `R = uxT`: `ux = u cosθ` = `30 cos60°` = `30 * 0.5` = `15 m/s`. `R = 15 5.196` = `77.94 meters`. --- 8.
1. Differentiation Strategies Group Work: Form mixed-ability groups where stronger students can explain concepts and assist weaker students.
Varied Tasks: Assign different levels of complexity in problems (e.g., simpler direct application of formula for some, multi-step problems requiring derivation or finding missing variables for others).
Peer Tutoring: Pair students with strong understanding with those who are struggling to explain concepts and assist with problem-solving. 8.
2. Remediation Strategies for Struggling Learners Focus on Components: Break down projectile motion into its independent horizontal and vertical components. Ensure mastery of basic kinematics equations for each component separately before combining.
Visual Aids: Use diagrams, trajectory sketches, and animations (if available) to visually represent the motion and the vector components of velocity and acceleration.
Step-by-Step Guidance: Provide explicit, step-by-step instructions for solving problems.
Use a checklist approach: Identify given variables (`u`, `θ`, `g`). Resolve `u` into `ux` and `uγ`. Identify the unknown quantity to be found (T, H, or R). Select the appropriate formula. Substitute values and calculate.
Simplified Problems: Start with problems where the angle of projection is `0°` (horizontal projection) or `90°` (vertical throw) to build confidence before introducing inclined projection.
Recap of Basic Kinematics: Review the SUVAT equations thoroughly as they are foundational to understanding projectile motion derivations. 8.
3. Extension Activities for High-Achieving Learners Advanced Problem Solving: Investigate projectile motion on an inclined plane. Solve problems where the landing point is at a different height than the projection point (e.g., throwing a stone from a cliff into a valley). Calculate the velocity (magnitude and direction) of a projectile at any given time or position during its flight.
Qualitative Analysis of Air Resistance: Discuss how air resistance affects the trajectory, range, and maximum height, leading to deviations from the ideal parabolic path. Encourage them to research or design a simple experiment to observe these effects (e.g., comparing the flight of a paper ball vs. a heavier, denser ball).
Explore Trajectory Equation: Challenge students to derive the equation of the trajectory (`y = x tanθ - (gx2) / (2u2 cos2θ)`) and use it to solve problems or graph projectile paths.
Research Project: Assign a mini-research project on historical projectile devices (e.g., catapults, trebuchets) or modern applications in sports science or military technology, focusing on the physics principles involved.
Projectile motion is not just a theoretical concept; it has numerous practical applications that resonate with the Nigerian context.
Sports and Athletics: Football (Soccer): The trajectory of a football when kicked for a goal, a long pass, or a free-kick. Players intuitively understand projectile motion to aim, control power, and consider the angle of kick to achieve desired outcomes (e.g., a high arc for a cross, a low powerful shot for a goal). Coaches use this understanding to train players in precision kicking.
Athletics: Events like javelin throw, shot put, and discus throw are classic examples. Athletes and their trainers constantly work on optimizing the initial velocity and angle of projection to maximize the range or height achieved. For instance, the optimal angle for maximum range in these events is slightly less than 45° due to factors like height of release and air resistance, but the fundamental principle of `θ=45°` for maximum range from ground level is a crucial starting point for understanding.
Agriculture and Irrigation: Water Sprinklers: In large Nigerian farms, especially for crops like rice, or in urban gardens, water sprinklers are used for irrigation. The design and placement of these sprinklers depend on understanding the range and height of the water jets (projectile motion) to ensure uniform and efficient water distribution without wastage. Engineers must calculate the required pressure and nozzle angle to achieve a specific coverage area.
Pumping water: When water is pumped from a bore-hole or river through a hosepipe, the trajectory of the water can be predicted using projectile principles to direct it accurately. Security and Military Applications (Ballistics): Artillery and Mortars: The trajectory of shells fired from artillery guns or mortars is a sophisticated application of projectile motion. Military strategists and engineers use advanced ballistic calculations to determine the exact angle and initial velocity required for a shell to hit a target at a specific range, taking into account factors like wind, air resistance, and Earth's rotation. This is crucial for national security and defense operations. ---