Lesson Notes By Weeks and Term v4 - SHS 3
DYNAMICS
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DYNAMICS
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Subject: Physics
Class: SHS 3
Term: 1st Term
Week: 14
Grade code: 3.1.3.LI.1
Strand code: 1
Sub-strand code: 3
Content standard code: 3.1.2.CS.1
Indicator code: 3.1.3.LI.1
Theme: MECHANICS AND MATTER
Subtheme: DYNAMICS
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This lesson bridges two fundamental concepts in Physics that you have already encountered: Newton's Third Law of Motion and the Principle of Conservation of Linear Momentum. We often see these principles in action without thinking about them. When a footballer like Mohammed Kudus kicks a ball, his foot exerts a force on the ball, but the ball also exerts a force back on his foot. When a market woman pushes a heavy cart, the cart pushes back on her. Today, we will go beyond simply stating these laws. We will use the mathematics of momentum to *prove* or *verify* that Newton's Third Law must be true during any interaction, like a collision.
This lesson builds on your prior knowledge. Let's start with a quick review and then connect the ideas. A. Review of Foundational Concepts Newton's Third Law of Motion: Statement: "For every action, there is an equal and opposite reaction." Meaning: If Body A exerts a force on Body B (the "action"), then Body B simultaneously exerts a force on Body A that is equal in magnitude and opposite in direction (the "reaction"). Crucial Point: The action and reaction forces act on *different* bodies. They never cancel each other out. Example: When pounding fufu, the pestle exerts a downward force on the fufu in the mortar (action). The fufu exerts an equal upward force on the pestle (reaction), which you can feel in your hands. Linear Momentum (p): Definition: The product of an object's mass and its velocity. It is a measure of the quantity of motion an object has. Formula: `p = mv` Units: kilogram-metre per second (kg m/s). Nature: Momentum is a vector quantity; it has both magnitude and direction. Impulse (J): Definition: The product of the force acting on an object and the time interval over which the force acts. Formula: `J = FΔt` Impulse-Momentum Theorem: The impulse applied to an object is equal to the change in the object's momentum. Derivation: From Newton's Second Law, `F = ma = m(v-u)/t`. Rearranging gives `Ft = m(v-u) = mv - mu = Δp`. Therefore: `Impulse = Change in Momentum` or `J = Δp`. B. The Principle of Conservation of Linear Momentum Statement: For a system of interacting objects, the total linear momentum remains constant, provided that no external net force acts on the system. Isolated System: A system on which the net external force is zero. In a collision between two balls on a smooth table, we can ignore friction and air resistance, making the balls an isolated system. The forces they exert on each other are *internal* forces. Mathematical Representation: For a two-body collision: Total momentum before collision = Total momentum after collision `m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂` Where: `m₁`, `m₂` = masses of body 1 and body 2 `u₁`, `u₂` = initial velocities of body 1 and body 2 `v₁`, `v₂` = final velocities of body 1 and body 2 C. Verifying Newton's Third Law from Momentum Change (The Core of the Lesson)
This is our main task today. We will start with the Principle of Conservation of Momentum and logically arrive at Newton's Third Law. This is a beautiful example of the consistency of physics.
Step-by-Step Derivation: Start with the conservation of momentum equation for a two-body collision in an isolated system: `m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂` Rearrange the equation to group the properties of each body together. Move all terms for body 1 to the left side and all terms for body 2 to the right side. `m₁v₁ - m₁u₁ = m₂u₂ - m₂v₂` Factor out the masses and recognise the change in momentum (Δp). `m₁(v₁ - u₁) = -(m₂v₂ - m₂u₂)` The term `m₁(v₁ - u₁)` is the change in momentum of body 1 (`Δp₁`). The term `m₂(v₂ - u₂)` is the change in momentum of body 2 (`Δp₂`). State the intermediate result: `Δp₁ = -Δp₂` This is a very important result! It states that during a collision, the change in momentum of one body is equal in magnitude and opposite in direction to the change in momentum of the other body. Introduce the time of interaction (Δt). The collision happens over a short period of time, `Δt`, which is the same for both bodies. Let's divide both sides of the equation by `Δt`. `Δp₁ / Δt = - (Δp₂ / Δt)` Apply Newton's Second Law in terms of momentum. We know that the average force `F` is the rate of change of momentum (`F = Δp / Δt`). `Δp₁ / Δt` is the average force exerted *on* body 1 by body 2. Let's call this `F₁`. `Δp₂ / Δt` is the average force exerted *on* body 2 by body 1. Let's call this `F₂`. Substitute the forces into the equation: `F₁ = -F₂`
Conclusion: This final equation is the mathematical statement of Newton's Third Law of Motion. It shows that the force on body 1 is equal in magnitude and opposite in direction to the force on body 2. We have successfully verified the Third Law using the principle of momentum conservation.