Lesson Note on Dynamics: Projectiles, Trajectory and Inclined Projection for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 3rd Term
Week: 8
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Dynamics
Previous Lesson: Dynamics: Connected Particles, Work, Energy, Power and Momentum.
Topic: DYNAMICS
Subject Matter: Projectiles, trajectory of projectiles, projection along inclined plane

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define projectile motion and state assumptions.
• Derive or recall the equations of motion for a projectile.
• Calculate the time of flight, maximum height, and horizontal range of a projectile.
• Explain the concept of trajectory and state its equation.
• Describe the approach to solving problems involving projection along an inclined plane.

Affective Domain:

• Appreciate the real-world applications of projectile motion.
• Show interest in solving problems related to projectiles.
• Demonstrate precision and accuracy in calculations.

Psychomotor Domain:

• Sketch the path (trajectory) of a projectile.
• Solve various numerical problems on projectile motion accurately.
• Apply the correct formulas and steps to solve problems involving inclined planes.

Social Domain:

• Collaborate with peers to discuss and solve projectile problems.
• Communicate mathematical ideas clearly when explaining solutions.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New Further Mathematics for Senior Secondary Schools by P.N. Okeke et al.
• Further Mathematics Project 2 by A.O. Kalejaiye et al.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing projectile motion paths and diagrams.
• Whiteboard and markers.
• Calculators.

Rationale for the Lesson

This lesson helps pupils understand the motion of objects launched into the air, which is fundamental in physics and engineering. It enables pupils to predict the path and landing points of objects, which has applications in sports, military, and various scientific fields.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of vectors, basic equations of motion (uniform acceleration), gravity, and resolution of forces from their previous studies in Mathematics and Physics.

Lesson Content/Board Summary

Projectile Motion

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 object is called a projectile, and its path is called its trajectory.

Assumptions in Projectile Motion

• Air resistance is negligible.
• The acceleration due to gravity (g) is constant in magnitude and direction.
• The Earth’s curvature is negligible.

Equations of Motion for a Projectile

Consider a projectile launched with initial velocity ‘u’ at an angle ‘θ’ to the horizontal. The motion can be resolved into horizontal and vertical components.

Horizontal Motion (constant velocity, a_x = 0):

• Velocity: V_x = u cos θ
• Displacement: x = (u cos θ)t

Vertical Motion (constant acceleration, a_y = -g):

• Initial Velocity: u_y = u sin θ
• Final Velocity: V_y = u sin θ – gt
• Displacement: y = (u sin θ)t – ½gt²
• V_y² = (u sin θ)² – 2gy

Key Parameters of Projectile Motion

1. Time of Flight (T): The total time the projectile remains in the air.

T = (2u sin θ) / g

2. Maximum Height (H): The greatest vertical distance reached by the projectile.

H = (u² sin² θ) / (2g)

3. Horizontal Range (R): The total horizontal distance covered by the projectile.

R = (u² sin 2θ) / g

Maximum range occurs when θ = 45°, giving R_max = u²/g.

Equation of Trajectory

The path of a projectile is a parabola. Its equation is given by:

y = x tan θ – (gx²) / (2u² cos² θ)

Worked Examples

Example 1: A stone is projected from the ground with an initial velocity of 20 m/s at an angle of 30° to the horizontal. Calculate:

a) The time of flight.

b) The maximum height reached.

c) The horizontal range.

(Take g = 10 m/s²)

Solution:

Given: u = 20 m/s, θ = 30°, g = 10 m/s²

Step 1: Calculate Time of Flight (T)

T = (2u sin θ) / g

T = (2 × 20 × sin 30°) / 10

T = (40 × 0.5) / 10

T = 20 / 10 = 2 seconds

Step 2: Calculate Maximum Height (H)

H = (u² sin² θ) / (2g)

H = (20² × (sin 30°)²) / (2 × 10)

H = (400 × (0.5)²) / 20

H = (400 × 0.25) / 20

H = 100 / 20 = 5 meters

Step 3: Calculate Horizontal Range (R)

R = (u² sin 2θ) / g

R = (20² × sin (2 × 30°)) / 10

R = (400 × sin 60°) / 10

R = (400 × 0.866) / 10

R = 346.4 / 10 = 34.64 meters

Example 2: A projectile is fired with an initial velocity of 50 m/s. If its horizontal range is 200 m, find the possible angles of projection. (Take g = 10 m/s²)

Solution:

Given: u = 50 m/s, R = 200 m, g = 10 m/s²

Step 1: Use the Range Formula

R = (u² sin 2θ) / g

200 = (50² × sin 2θ) / 10

Step 2: Solve for sin 2θ

200 × 10 = 2500 × sin 2θ

2000 = 2500 sin 2θ

sin 2θ = 2000 / 2500

sin 2θ = 0.8

Step 3: Find 2θ

2θ = arcsin(0.8)

2θ ≈ 53.13°

Step 4: Find the first angle θ

θ = 53.13° / 2 ≈ 26.57°

Step 5: Find the second possible angle (using sin(180° – x) = sin x)

Another value for 2θ is 180° – 53.13° = 126.87°

2θ = 126.87°

θ = 126.87° / 2 ≈ 63.43°

The possible angles of projection are approximately 26.57° and 63.43°.

Projection Along an Inclined Plane

When a projectile is launched from or lands on an inclined plane, the analysis involves rotating the coordinate system such that the x-axis is along the incline and the y-axis is perpendicular to it. Gravity (g) must then be resolved into components along these new axes.

• The component of gravity parallel to the incline is g sin α (where α is the angle of inclination).
• The component of gravity perpendicular to the incline is g cos α.
• The equations of motion are applied using these resolved components of gravity.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher asks pupils what happens when a ball is kicked or a stone is thrown. The teacher then guides pupils to recall their knowledge of motion under gravity.
Pupils’ Activity: Pupils respond by describing the path of the ball/stone and recall concepts of velocity and gravity.
Learning Point: Pupils are introduced to the concept of projectile motion and its relevance.

Step 2: Explanation of Projectile Motion and Assumptions

Time: 8 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines projectile motion, explains the terms ‘projectile’ and ‘trajectory’, and states the key assumptions made in its study (negligible air resistance, constant gravity).
Pupils’ Activity: Pupils listen attentively, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and basic conditions for projectile motion.

Step 3: Derivation/Statement of Equations of Motion

Time: 10 minutes
Teaching Skill: Derivation/Problem Solving
Teacher’s Activity: The teacher guides pupils to resolve initial velocity into horizontal and vertical components. The teacher then states and explains the formulas for time of flight, maximum height, and horizontal range, linking them to the general equations of motion.
Pupils’ Activity: Pupils observe, copy the formulas, and participate in brief discussions on how the components relate to the overall motion.
Learning Point: Pupils learn the fundamental equations for analyzing projectile motion.

Step 4: Worked Examples on Projectile Motion

Time: 10 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher solves Example 1 and Example 2 from the board summary, demonstrating step-by-step application of the formulas for time of flight, maximum height, and range. The teacher emphasizes units and clear presentation.
Pupils’ Activity: Pupils follow the steps, ask questions, and attempt to solve similar problems or confirm the calculations.
Learning Point: Pupils develop problem-solving skills for standard projectile motion problems.

Step 5: Trajectory of a Projectile

Time: 5 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher explains that the path of a projectile is parabolic and presents the equation of trajectory, explaining its significance in describing the path.
Pupils’ Activity: Pupils listen, copy the equation, and understand the shape of the projectile’s path.
Learning Point: Pupils understand the mathematical representation of a projectile’s path.

Step 6: Projection Along an Inclined Plane

Time: 5 minutes
Teaching Skill: Conceptual Explanation
Teacher’s Activity: The teacher introduces the concept of projection along an inclined plane, explaining how the coordinate system is rotated and how gravity is resolved into components along the new axes.
Pupils’ Activity: Pupils listen and understand the modification required for inclined plane problems.
Learning Point: Pupils grasp the foundational concept for analyzing projectiles on inclined surfaces.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define projectile motion.
2. State any two assumptions made in projectile motion.
3. Write down the formula for the horizontal range of a projectile.
4. A stone is thrown with an initial velocity of 30 m/s at an angle of 60° to the horizontal. Calculate its time of flight. (Take g = 10 m/s²)

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the definitions, formulas, and problem-solving approaches for projectiles. The teacher then assigns homework.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their understanding and prepare for further practice.

Lesson Keywords

• Projectile – An object thrown into the air that is only subject to gravity.
• Trajectory – The path followed by a projectile.
• Range – The horizontal distance covered by a projectile.
• Time of Flight – The total time a projectile spends in the air.
• Maximum Height – The highest vertical point reached by a projectile.
• Inclined Plane – A flat surface set at an angle to the horizontal.

Differentiation

For pupils who grasp concepts quickly, the teacher can introduce more complex problems, such as finding the angle of projection given range and time, or problems involving relative motion. For pupils needing more support, the teacher will provide simplified examples and extra guidance during problem-solving, focusing on one formula at a time and providing step-by-step solutions.

Note for teachers using this lesson plan

Ensure that pupils have a strong grasp of vector resolution and the basic equations of motion before starting this topic. Emphasize the importance of resolving initial velocity and gravity components correctly. Encourage pupils to draw diagrams for each problem, especially for inclined plane scenarios, to visualize the forces and motion. Provide ample practice problems to reinforce understanding and build confidence in solving numerical questions.