Class 11 > Unit # 03: Dynamics > Momentum & Law of Conservation of Momentum

 

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Unit # 3: Dynamics — Momentum and Conservation of Momentum

Lecture Overview

This lecture from Talha's Physics Academy covers linear momentum, its formulas and units, impulse, Newton's second law in terms of momentum, and the complete proof of the law of conservation of momentum for Class 11 Physics (Unit 3: Dynamics).

Source Video: Momentum and Law of Conservation of Momentum - Unit 3 Dynamics - #TP11 29

1. Momentum

Definition

Momentum can be defined physically as the quantity of motion contained in a body, and mathematically as the product of the mass and velocity of a body.

$$\mathbf{P} = m\mathbf{V}$$

Where:

• \(\mathbf{P}\) = Momentum (Vector quantity)
• \(m\) = Mass of the body
• \(\mathbf{V}\) = Velocity of the body

Nature & Units

Momentum is a vector quantity, and its direction is parallel to the direction of velocity. Its S.I. unit is \(\text{kg}\cdot\text{m/s}\) or \(\text{N}\cdot\text{s}\) (Newton-Second).

2. Newton’s Second Law and Linear Momentum

Newton's Second Law can be expressed in terms of momentum to show that the time rate of change of an object's momentum is equal to the net constant force acting on it:

$$\mathbf{F} = \frac{\Delta \mathbf{P}}{t} = \frac{m\mathbf{v}_f - m\mathbf{v}_i}{t}$$

3. Impulse

Definition

The impulse of a force is the product of the average force and the time interval \(\Delta t\) during which it acts.

$$\text{Impulse} = \mathbf{F} \times \Delta t = \Delta \mathbf{P}$$

4. Law of Conservation of Momentum

Statement

“The total momentum of an isolated system before collision is always equal to total momentum after collision.”

Explanation & Mathematical Proof

\

Consider an isolated system consisting of two interacting bodies A and B of masses \(m_1\) and \(m_2\) moving with initial velocities \(u_1\) and \(u_2\). After colliding, they move with velocities \(v_1\) and \(v_2\).

• Total momentum before collision: \(m_1u_1 + m_2u_2\)
• Total momentum after collision: \(m_1v_1 + m_2v_2\)

During collision for a time interval \(t\), body A exerts a force on body B (\(\mathbf{F}_{\text{A on B}}\)), and by Newton's Third Law, body B exerts an equal and opposite force on body A (\(\mathbf{F}_{\text{B on A}}\)):

$$\frac{m_1v_1 - m_1u_1}{t} = -\left(\frac{m_2v_2 - m_2u_2}{t}\right)$$

Multiplying both sides by \(t\) and simplifying yields the conservation equation:

$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$

This proves that the total momentum of an isolated system remains constant.

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