Conservation of Linear Momentum – Complete Notes for B.Sc. 1st Year Physics Students || Bsc Physics notes 1st year

Conservation of Linear Momentum – Complete Notes for B.Sc. 1st Year Physics Students

🔹 Introduction

In the study of mechanics, momentum is one of the most fundamental physical quantities. It describes the quantity of motion that an object possesses and plays a vital role in understanding how forces influence motion. Among the key principles in classical mechanics is the law of conservation of linear momentum, which is not only essential for solving problems in physics but also helps explain a wide range of real-life phenomena—from the recoil of guns to rocket launches, car collisions, and more.

This set of notes is specially designed for B.Sc. 1st Year Physics students to help you understand this concept thoroughly and apply it confidently in exams and problem-solving.

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🔹 What Is Linear Momentum?

Linear momentum (also just called momentum) is a vector quantity that represents the product of an object’s mass and velocity.

$\vec{p} = m \vec{v}$

Where:

• $\vec{p}$: linear momentum

• $m$: mass of the object

• $\vec{v}$: velocity of the object

Its SI unit is kg·m/s.

Momentum has both magnitude and direction. If either mass or velocity increases, momentum increases.

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🔹 Statement of the Law of Conservation of Linear Momentum

"If no external force acts on a system of particles, the total linear momentum of the system remains constant."

This means that when objects interact (like during a collision), their total momentum before the interaction equals the total momentum after the interaction—provided no external force disturbs the system.

Mathematically:

$\text{Total initial momentum} = \text{Total final momentum}$

Or,
$\sum m_i \vec{u}_i = \sum m_i \vec{v}_i$

Where:

• $m_i$: mass of the i-th particle

• $\vec{u}_i$: initial velocity

• $\vec{v}_i$: final velocity

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🔹 Derivation Using Newton’s Laws

Let’s derive the law for two interacting bodies:

Suppose two bodies A and B, with masses $m_1$ and $m_2$, are moving with initial velocities $u_1$ and $u_2$. They interact (e.g., collide), and after the interaction, their velocities become $v_1$ and $v_2$.

According to Newton's Third Law:
$F_{AB} = -F_{BA}$

Let the time of interaction be $t$. Then, using Newton's Second Law:
$F = \frac{\Delta p}{\Delta t}$

So,
$\frac{m_1 v_1 - m_1 u_1}{t} = -\frac{m_2 v_2 - m_2 u_2}{t}$

Multiply both sides by $t$:
$m_1 v_1 - m_1 u_1 = - (m_2 v_2 - m_2 u_2)$

Rearranging:
$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$

Hence, momentum is conserved.

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🔹 Conditions for Momentum Conservation

To apply this law accurately, the following conditions must be satisfied:

1. Isolated System: There should be no net external force acting on the system.

2. Internal Forces Only: The forces acting should be mutual forces between the particles.

3. Short Time Interval: The interaction time is generally considered short (especially in collision problems).

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🔹 Applications of Conservation of Momentum

1. Recoil of a Gun:

   • When a bullet is fired, the gun moves backward. This backward motion is due to the conservation of momentum.

   • Let $m$ be mass of bullet, $M$ be mass of gun, $v$ is bullet’s velocity, and $V$ is gun’s recoil velocity.

   • Since initial momentum is 0:
     $mv + MV = 0 \Rightarrow V = -\frac{mv}{M}$

2. Rocket Propulsion:

   • Rockets expel gases in the opposite direction of motion.

   • This backward force causes the rocket to move forward — conserving momentum.

3. Collisions:

   • In elastic collisions, both momentum and kinetic energy are conserved.

   • In inelastic collisions, only momentum is conserved.

4. Explosion Problems:

   • In an explosion, parts fly off in different directions, but the total momentum of the system remains conserved.

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🔹 Elastic vs Inelastic Collisions (Based on Momentum)

Feature	Elastic Collision	Inelastic Collision
Momentum Conservation	✅ Yes	✅ Yes
Kinetic Energy Conservation	✅ Yes	❌ No
Real-life Example	Billiard balls	Car crash

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🔹 Vector Nature of Momentum

Momentum is a vector. This means:

• Direction matters.

• You must consider x, y, and z components (especially in 2D/3D problems).

In two-body problems:
$\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$

i.e.,
$m_1 \vec{u}_1 + m_2 \vec{u}_2 = m_1 \vec{v}_1 + m_2 \vec{v}_2$

You may need to resolve velocities into components and apply conservation separately in each direction.

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🔹 Important Example Questions

Q1: A 10 kg gun fires a 0.5 kg bullet with a speed of 400 m/s. Find the recoil velocity of the gun.

Solution:
$mv + MV = 0 \Rightarrow V = -\frac{mv}{M} = -\frac{0.5 \times 400}{10} = -20\,\text{m/s}$

Q2: Two ice skaters push off from each other. One has mass 50 kg and moves at 3 m/s. What is the velocity of the second skater if they were initially at rest?

Solution:
Let second skater’s mass be 75 kg, velocity = $v$
$50 \times 3 + 75 \times v = 0 \Rightarrow v = -2\,\text{m/s}$

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🔹 Real-Life Examples

1. Jumping from a Boat: You move forward; the boat moves backward.

2. Airplane Jet Engines: Backward thrust of gases moves the airplane forward.

3. Atomic Reactions: Particles emitted in nuclear reactions conserve total momentum.

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🔹 Conservation of Momentum in Two Dimensions

In cases like glancing collisions, momentum must be conserved in both x and y directions.

• Use:
  $\sum p_x^{\text{initial}} = \sum p_x^{\text{final}}$
  $\sum p_y^{\text{initial}} = \sum p_y^{\text{final}}$

This is often visualized using vector diagrams or trigonometry.

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🔹 Summary – Key Points to Remember

• Momentum = mass × velocity (vector quantity)

• Conserved in isolated systems (no external force)

• Applies to collisions, explosions, and propulsion systems

• Derived from Newton’s 3rd law and 2nd law

• In 2D problems, apply separately in x and y directions

• Elastic collision: KE conserved; Inelastic: KE not conserved

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🔹 Final Words

Understanding the law of conservation of linear momentum is essential for every physics student. It's not only a powerful tool for solving mechanics problems but also a principle that governs many natural and man-made phenomena. Whether you're solving numerical problems, analyzing experiments, or preparing for exams, this concept will frequently come up in your academic journey.

Always look for how momentum changes and remember: if there’s no external force, momentum is your best friend!

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Conservation of Linear Momentum 

Important notes for BSC 1st year Physics students