Linear Momentum of a System

Linear momentum of a system is a fundamental concept in physics that describes the combined motion of all objects or particles within a system. It is defined as the vector sum of the momenta of all individual components. Since momentum is the product of mass and velocity, this idea extends to systems with multiple moving or interacting bodies. Understanding system momentum is essential for analyzing collisions, explosions, and other dynamic events. A key related principle is the law of conservation of momentum, which states that the total momentum of a closed system remains constant if no external forces act on it. This principle is widely applied in physics, engineering, and astrophysics to explain and predict motion in both everyday and cosmic systems.

1.0Definition of Linear Momentum

Linear momentum of an object is the product of its mass and its velocity. The linear momentum (p) of a particle is defined as:

$p=mv$

Where m = mass of the particle ,v = velocity vector of the particle

• Linear momentum is a vector quantity.
• Its SI unit is kg·m/s.
• Momentum depends on both mass and velocity; doubling either doubles momentum.

2.0Linear Momentum of a System of Particles

Consider a system of n particles with masses $m_1,m_2,\dots ,m_n$ and velocities ${\vec{v}}_1,{\vec{v}}_2,\dots ,{\vec{v}}_n$

These particles may interact and can also be influenced by external forces.

For a system of n particles, the total linear momentum is the vector sum of the momenta of all individual particles:

$\vec{P}={\sum }_{i=1}^n{m}_i{\vec{v}}_i$

Where $m_i$ = mass of the ith particle , $v_i$ = velocity of the ith particle

If M is the total mass of the system and $\vec{V}$ is the velocity of its center of mass then

$\vec{P}=M\vec{V}$

Key Points

The total linear momentum of a system equals the product of the system’s total mass and the velocity of its center of mass.

Differentiating with respect to time

$\frac{d\vec{P}}{dt}=M\frac{d\vec{V}}{dt}=M\vec{A}={\vec{F}}_{\text{ext}}$

This is an extension of Newton’s second law for a system of particles.

For an isolated system (no external forces): $\frac{d\vec{P}}{dt}=0\Rightarrow \vec{P}=\text{constant}\Rightarrow \vec{V}=\text{constant}$

• The velocity of the center of mass remains constant if no external forces act on the system.
• Internal forces (forces particles exert on each other) do not affect the motion of the center of mass.
• If the center of mass is at rest and no external forces act, it will remain at rest.

3.0Principle of Conservation of Linear Momentum

The law of conservation of linear momentum states that the total linear momentum of an isolated system of particles remains constant if no external force acts on the system.

Mathematically: $P_{\text{initial}}=P_{\text{final}}$

4.0Mathematical Formulation of Momentum in a System

For two particles with masses $m_{1}and{m}_2$  and velocities $v_{1}and{v}_2$

$P_{\text{total}}=m_1{v}_1+m_2{v}_2$

After interaction, if external forces are absent:

$m_1{v}_1+m_2{v}_2=m_1{v}_1^{\prime }+m_2{v}_2^{\prime }$

For perfectly inelastic collisions:

$v_{\text{final}}=\frac{m_1{v}_1+m_2{v}_2}{m_1+m_2}$

5.0Center of Mass and Linear Momentum

The center of mass (COM) of a system is the weighted average position of all particles: $R_{\text{COM}}=\frac{\sum m_i{r}_i}{\sum m_i}$

The velocity of the center of mass: $V_{\text{COM}}=\frac{\sum m_i{v}_i}{\sum m_i}$

Related with momentum:

$P_{\text{system}}=M_{\text{total}}{V}_{\text{COM}}$

Where $M_{\text{total}}=\sum m_i$

6.0Collisions and Momentum Transfer in Systems

(a) Elastic Collisions

• Both momentum and kinetic energy are conserved.
• Example: Two billiard balls colliding.
• $v_1^{\prime }=\frac{(m_1-m_2)v_1+2m_2{v}_2}{m_1+m_2},v_2^{\prime }=\frac{(m_2-m_1)v_2+2m_1{v}_1}{m_1+m_2}$

(b) Inelastic Collisions

• Momentum conserved, kinetic energy not conserved.
• Example: Clay balls sticking together after collision.
• Perfectly inelastic case
• $v_{\text{final}}=\frac{m_1{v}_1+m_2{v}_2}{m_1+m_2}$

7.0Impulse and Momentum Change in a System

Impulse is the product of the force applied on an object and the time duration for which the force is applied.

It is a vector quantity and is equal to the change in momentum of the object.

$J=\Delta P=F_{\text{ext}}\Delta t$

Where $F_{ext}$=external force, t= time interval of force

8.0Applications of Linear Momentum in Real Life

• Vehicle collisions: Momentum conservation predicts post-collision velocities.
• Rocket propulsion: Mass ejection changes system momentum.
• Ball games: Transfer of momentum during collisions in billiards, cricket, or football.
• Recoil of firearms: Momentum conservation explains backward motion of guns.
• Spacecraft docking and satellites: Momentum principles govern motion and docking maneuvers.

9.0Sample Illustration Questions on Linear Momentum of a System

Illustration-1: Two blocks of masses 2 kg and 3 kg move toward each other with velocities 4 m/s  and 2 m/s . They collide and stick together. Find final velocity.

Solution:

$v_{\text{final}}=\frac{2(4)+3(-2)}{2+3}=\frac{8-6}{5}=0.4\text{m/s}$

Illustration- 2: A 0.05 kg bullet is fired at 200 m/s from a 5 kg gun. Find recoil velocity of the gun.

Solution:

$m_{\text{gun}}{v}_{\text{gun}}=m_{\text{bullet}}{v}_{\text{bullet}}{v}_{\text{gun}}=\frac{0.05\times 200}{5}=2\text{m/s}$

Illustration- 3: A 2 kg ball moving at 3 m/s collides elastically with a 1 kg stationary ball. Find velocities after collision in one dimension.

Solution:

$v_1^{\prime }=\frac{(2-1)3+2(1.0)}{2+1}=1\text{m/s}$

$v_2^{\prime }=\frac{(1-2)0+2(2.3)}{2+1}=4\text{m/s}$