6.4 Linear Momentum of a System of Particles – Class 11 Physics

When dealing with systems consisting of many particles, each particle may move differently and experience different forces. Despite this complexity, a remarkably simple and powerful quantity governs the overall translational behaviour of the system — linear momentum.

Linear momentum not only connects directly with Newton's laws, but also leads to one of the most important conservation principles in physics. In competitive examinations like JEE Main and JEE Advanced, conservation of momentum forms the backbone of collision problems, recoil systems, explosions, and multi-body interactions.

This section develops the complete theory of linear momentum for a system of particles and establishes its deep connection with centre of mass motion.

1. Linear Momentum of a Single Particle

For a single particle of mass moving with velocity , linear momentum is defined as:

Momentum is a vector quantity and has the same direction as velocity.

Newton's Second Law can be written in momentum form as:

This form becomes especially useful for systems of particles.

2. Total Linear Momentum of a System

Consider a system of particles with masses and velocities .

The total linear momentum of the system is defined as the vector sum of the individual momenta:

Let the total mass of the system be:

The velocity of the centre of mass is given by:

Multiplying both sides by gives:

Thus, we obtain the important relation:

This equation states that the total linear momentum of a system is equal to the momentum of a single particle of mass moving with velocity equal to the centre of mass velocity.

This result links momentum conservation directly to centre of mass motion.

3. Time Rate of Change of Total Momentum

To understand how momentum evolves, we differentiate total momentum with respect to time:

Applying Newton's Second Law to each particle:

Here:

• represents external forces acting on particles .
• represents internal forces due to interactions with other particles.

Summing over all particles:

By Newton's Third Law, internal forces occur in equal and opposite pairs. Therefore,

Thus, we obtain:

This equation is extremely important. It states that the rate of change of total linear momentum of a system equals the net external force acting on the system.

4. Conservation of Linear Momentum

If no external force acts on the system:

Then,

Hence,

This is the Law of Conservation of Linear Momentum.

Using the relation , if total mass remains constant:

Thus, in absence of external force, the centre of mass moves with uniform velocity.

This principle holds even if the particles undergo violent internal interactions.

5. Impulse–Momentum Theorem

From the equation:

Integrating over time interval from to :

The quantity

is called impulse.

Thus,

This relation plays a crucial role in analyzing collisions, especially when forces act for very short durations.

6. Applications in JEE-Level Problems

6.1 Recoil of Gun

If a gun of mass fires a bullet of mass with velocity and the system was initially at rest:

Therefore,

Total momentum before and after firing remains zero.

6.2 Explosion of a Body Initially at Rest

If a body initially at rest explodes into fragments:

The vector sum of momenta of all fragments must be zero.

6.3 Two-Particle System with Internal Interaction

If two particles interact only through internal forces and no external force acts:

Their individual velocities may change, but total momentum remains unchanged.

6.4 Centre of Mass Frame Simplification

In the centre of mass frame:

This greatly simplifies elastic collision derivations and is heavily used in advanced problems.

6.5 Momentum Conservation in Collisions

Even when kinetic energy is not conserved (inelastic collision), momentum conservation still holds if no external force acts.

This distinction between momentum conservation and energy conservation is very important in competitive exams.

7. Deeper Conceptual Insights

• Momentum conservation arises from Newton's laws.
• Internal forces cannot alter total momentum.
• Momentum conservation is valid in all types of collisions (elastic and inelastic).
• The equation directly connects momentum with the centre of mass motion.
• Momentum conservation often allows solving problems without calculating forces explicitly.

Understanding these principles allows solving multi-body systems using short and elegant reasoning.

FAQs

Q1. What is total linear momentum of a system?

It is the vector sum of individual particle momenta.

Q2. Why do internal forces not affect total momentum?

Because internal forces cancel in equal and opposite pairs according to Newton's Third Law.

Q3. When is linear momentum conserved?

When the net external force is zero, i.e., .

Q4. How is momentum related to centre of mass velocity?

Through the equation .

Q5. Is momentum conserved in explosions and collisions?

Yes, provided no external force acts during the interaction.

Conclusion

Linear momentum of a system of particles provides a unifying principle for understanding collisions, explosions, recoil, and multi-body interactions. The fundamental relation

connects momentum directly to centre of mass motion. When the net external force is zero, conservation of momentum becomes one of the most powerful analytical tools in mechanics.

A strong command of this concept is essential before progressing to angular momentum and rotational dynamics in the later sections of Chapter 6.