What Is the Conservation of Momentum?

The conservation of momentum is a fundamental concept in physics describing how the total momentum of a system remains constant if no external force acts upon it. This principle applies to collisions, explosions, and various other mechanical interactions, and is essential for solving problems involving isolated systems in classical mechanics.

Definition of Momentum

Momentum is a physical quantity defined for a moving object as the product of its mass and velocity. It is a vector quantity, possessing both magnitude and direction, and is mathematically expressed as $ \vec{p} = m \vec{v} $, where $m$ is the mass and $\vec{v}$ is the velocity.

The study of momentum provides a foundation for understanding the forces required to accelerate or stop an object. This concept is further elaborated in Momentum Explained.

Law of Conservation of Momentum

The law of conservation of momentum states that for an isolated system, the vector sum of the momenta of all objects remains constant, provided no net external force acts on the system. Internal forces resulting from interactions within the system do not affect the total momentum.

If two objects, labeled $A$ and $B$, interact with each other, the forces they exert are equal in magnitude and opposite in direction, according to Newton's third law. If $ \vec{F}_{AB} $ acts on $A$ due to $B$, then $ \vec{F}_{BA} = -\vec{F}_{AB} $ acts on $B$ due to $A$.

Mathematical Formulation of Conservation of Momentum

Consider two objects with masses $ m_1 $ and $ m_2 $, and velocities $ \vec{u}_1 $ and $ \vec{u}_2 $ before interaction, and velocities $ \vec{v}_1 $ and $ \vec{v}_2 $ after interaction. According to the law:

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

This equation indicates that the total momentum of the system before collision equals the total momentum after collision, assuming the system is isolated. Such formulations are detailed further in Conservation of Momentum in Explosions.

Impulse and Change in Momentum

When two bodies collide, they exert equal and opposite forces on each other for the same duration. The impulse experienced by each is the product of force and collision time, which is equal in magnitude and opposite in direction for both bodies.

Impulse is also equal to the change in momentum, resulting in $ m_1 \Delta \vec{v}_1 = - m_2 \Delta \vec{v}_2 $. Therefore, each object experiences an equal and opposite change in momentum, preserving the total momentum of the system.

Generalization to n-Body Systems

For a system with $n$ interacting bodies, all internal forces cancel in pairs due to Newton’s third law. The general law for an isolated system is: $\sum\limits_{i=1}^n m_i \Delta \vec{v}_i = 0$. This vector sum shows that any changes in individual momenta must sum to zero.

The principle applies to a range of physical phenomena, including multiparticle interactions, as described in Conservation of Momentum Revision Notes.

Types of Collisions and Momentum Conservation

Momentum is always conserved in an isolated system, regardless of whether collisions are elastic or inelastic. In elastic collisions, both kinetic energy and momentum are conserved. In inelastic collisions, only momentum is conserved; kinetic energy may change.

• Momentum conservation holds for both elastic and inelastic collisions
• Internal forces involved are equal and opposite

Conservation of Momentum in Explosions

In an explosion, a single body splits into several fragments due to internal forces. If the initial momentum of the body is zero, the vector sum of the momenta of all fragments after the explosion must also be zero, satisfying conservation of momentum.

For a bomb of mass $M$ at rest, splitting into $n$ fragments, $\sum\limits_{i=1}^n m_i \vec{v}_i = 0$. The application to explosive events is outlined in Conservation of Momentum in Explosions.

Illustrative Example: Collision Between Two Objects

Suppose a 120 kg body moves at 2 m/s east and an 80 kg body moves at 8 m/s west. After a head-on collision, both bodies stick together and move as one unit. The total momentum before and after collision remains constant, confirming momentum conservation.

Applications of Conservation of Momentum

Conservation of momentum has practical significance in rockets, firearms, and motor propulsion systems. In rockets, the ejection of exhaust gases generates momentum in the opposite direction, causing the rocket to accelerate upward.

In firearms, firing a bullet results in the gun recoiling backward. Both the forward momentum of the bullet and the backward momentum of the gun sum to zero, illustrating momentum conservation. Additional examples are explored in Applications of Conservation of Momentum.

• Rocket propulsion by conservation of momentum
• Recoil of firearms upon firing
• Motorboats advancing as water is pushed backward

Conservation of Momentum in Fluids

In fluid mechanics, the conservation of momentum plays a crucial role in fluid flow analysis. The momentum equation accounts for variations in velocity and mass flow in a control volume, ensuring that the change in momentum equals the net external force acting on the system.

This formulation allows determination of fluid forces in pipes, jets, and nozzles, forming the basis for engineering calculations.

Mathematical Equation for Conservation of Momentum

The general equation for conservation of linear momentum for a system can be written as:

$\dfrac{d\vec{P}}{dt} = \vec{F}_{\text{ext}}$

For $\vec{F}_{\text{ext}} = 0$, then $\vec{P}$ is constant, where $\vec{P}$ is the total momentum of the system. Further discussion relating momentum to work and energy may be found in Work, Energy, and Power Relation.

Key Points on Conservation of Momentum

• Holds true for all isolated systems
• Independent of the type of internal forces
• Applicable to collisions, explosions, and fluid systems

Summary

The conservation of momentum is a fundamental law in physics, relevant in collisions, explosions, and many real-world applications. All physical systems follow this law under the absence of external forces.