3.7 The Law of Conservation of Linear Momentum

Introduction

The Law of Conservation of Linear Momentum is a fundamental principle in physics. It states that the total momentum of an isolated system remains constant. Within a system, momentum can be transferred from one object to another, but the total amount of momentum never changes. This principle is essential for analyzing the dynamics of collisions, explosions, and rocket propulsion.

Statement of the Law

The Law of Conservation of Linear Momentum states:

The Isolated System

This law applies only to an isolated system, which is a collection of objects that do not interact with anything external to the system. In practice, this means that external forces such as friction or air resistance are negligible compared to the internal forces between colliding objects.

Mathematical Formulation

For an isolated system, the total initial momentum (${\vec{p}}_{initial}$) equals the total final momentum (${\vec{p}}_{final}$):

${\vec{p}}_{initial}={\vec{p}}_{final}$

This implies that the change in the system's total momentum is zero:

$\Delta {\vec{p}}_{system}=0$

For a system of two colliding bodies with masses $m_1$ and $m_2$:

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

Where:

• ${\vec{v}}_1,{\vec{v}}_2$ represent initial velocities
• ${\vec{v}}_1^{\prime },{\vec{v}}_2^{\prime }$ represent final velocities

Derivation from Newton's Laws

The law of conservation of momentum is a direct consequence of Newton's Second and Third Laws. Consider two particles, $m_1$ and $m_2$, that collide.

1. During the collision, $m_1$ exerts a force ${\vec{F}}_{21}$ on $m_2$.
2. By Newton's Third Law, $m_2$ exerts an equal and opposite force ${\vec{F}}_{12}$ on $m_1$: ${\vec{F}}_{12}=-{\vec{F}}_{21}$.
3. From Newton's Second Law, force is the rate of change of momentum: $\vec{F}=\frac{\Delta \vec{p}}{\Delta t}$
4. Therefore: $\frac{\Delta {\vec{p}}_1}{\Delta t}=-\frac{\Delta {\vec{p}}_2}{\Delta t}$
5. This simplifies to $\Delta {\vec{p}}_1+\Delta {\vec{p}}_2=0$, meaning the total change in momentum of the system is zero.

Applications of Momentum Conservation

1. Collisions

In any collision (elastic or inelastic), as long as the system is isolated, the total momentum before the collision equals the total momentum after. This is the primary principle used to calculate the velocities of objects after they collide.

For elastic collisions in one dimension, the relative speed of approach before collision equals the relative speed of separation after collision. See Elastic And Inelastic Collision→ for detailed treatment.

2. Explosions

When an object at rest explodes, its initial momentum is zero. To conserve momentum, the vector sum of the momenta of all fragments must also be zero. The fragments fly off in opposite directions.

Example: A Bomb at Rest

• Initial Momentum: ${\vec{p}}_i=0$
• If it splits into two fragments, $A$ and $B$:
• Final Momentum: ${\vec{p}}_f=m_A{\vec{v}}_A+m_B{\vec{v}}_B=0$
• This implies: $m_A{\vec{v}}_A=-m_B{\vec{v}}_B$

The fragments have equal and opposite momenta.

3. Recoil of a Gun

This is a classic example of momentum conservation during an explosion.

• Before Firing: The total momentum of the gun and bullet is zero.
• After Firing: The bullet moves forward with momentum $p_{bullet}$, and the gun moves backward with equal and opposite momentum ($-p_{gun}$) to keep total momentum zero:

$m_{gun}{\vec{v}}_{gun}=-m_{bullet}{\vec{v}}_{bullet}$

4. Rocket and Jet Propulsion

A rocket in space is an isolated system. To move forward, it expels hot gases backward at high velocity. The rocket gains forward momentum equal in magnitude to the backward momentum of the ejected gases, conserving total momentum of the system (rocket + fuel).

Possible Questions and Answers

Q: Is momentum conserved when a ball is dropped and bounces off the floor? A: If the system is just the ball, momentum is not conserved because an external force (gravity from the Earth) acts on it. However, if the system is defined as the ball + Earth, then momentum is conserved. The Earth recoils very slightly when the ball falls and bounces.

Q: What is the difference between conservation of momentum and conservation of energy? A: Momentum is a vector, and its conservation is a fundamental law that applies to all interactions in an isolated system. Kinetic energy is a scalar and is only conserved in elastic collisions. In inelastic collisions, momentum is still conserved, but kinetic energy is not.