Conservation of Linear Momentum

Conservation of linear momentum is a key principle governing interacting objects’ behavior in various physical scenarios. It states that the momentum before an event must equal the momentum after the event, provided there are no external forces involved. The conservation of linear momentum finds wide application across various fields of physics and engineering. It explains phenomena such as collisions, explosions, and the motion of celestial bodies. In this article, we’ll talk about the conservation of linear momentum, its formula, and how it works in real life.

What is Conservation of Linear Momentum?

Conservation of linear momentum is a fundamental principle in physics, stating that the total momentum of an isolated system remains constant if no external forces act on it. In other words, if no external forces are exerted on a system, the system’s total momentum before a particular event must be equal to the total momentum after that event. This principle is derived from Newton’s laws of motion and applies to a wide range of physical phenomena.

Conservation of Linear Momentum Formula

The formula for the conservation of linear momentum states that the total linear momentum of an isolated system remains constant if no external forces act on it. Mathematically, it can be expressed as:

∑p_{initial =} ∑p_final

where:

• ∑p_initial is the total initial momentum of the system,
• ∑p_final is the total final momentum of the system.

This equation means that the sum of the momenta of all objects in the system before an event is equal to the sum of their momenta after the event. It reflects the conservation principle that momentum is conserved in a closed system where no external forces are acting.

What is Linear Momentum

Linear momentum, often simply referred to as momentum, is a fundamental concept in physics that describes the quantity of motion possessed by an object. It is a vector quantity, meaning it has both magnitude and direction. Linear Momentum quantifies the amount of motion an object possesses, objects with larger masses or higher velocities have greater linear momentum. According to Newton’s laws of motion, an external force acting on an object can change its momentum, resulting in acceleration or deceleration.

Linear Momentum Formula

The formula of linear momentum represents the product of an object’s mass and its velocity. Mathematically it can be expressed as:

p = m․v

where:

• p is the linear momentum
• m is the mass of the object
• v is the velocity of the object

Dimensional Formula of Linear Momentum

The dimensional formula of Linear momentum is:

[MLT^-1 ]

where:

• M is the mass of the object
• L is the length
• T is the time

Conservation of Linear Momentum Equation

Let us consider a system comprising of n particles labelled from 1 to n each with masses m₁, m_2,………….m_n and velocities v₁, v₂,……………v_n

The total linear momentum of the system is given by the sum of individual momenta of all particles:

p = m₁v₁ + m₂v₂ + …………… + m_nv_n

p = p₁ + p₂ + ………….. + p_n

According to Newton’s Second Law of Motion,

F = ma

⇒F= mdv/dt

⇒F= d(mv)/dt

⇒ F= dp/dt

For an isolated system where no external force is present (F=0), the rate of change of momentum will also be zero (dp/dt=0). This implies:

p = constant

Therefore, if no external force is exerted, the system’s linear momentum remains constant, implying that the sum of the momenta of all particles within the system remains constant.

Conservation of Linear Momentum Applications

Applications of the conservation of linear momentum are widespread across various fields of science and engineering. Some of them are:

• Collision Analysis: Conservation of linear momentum is extensively used in analyzing collisions between objects, whether they are elastic or inelastic collisions. This principle helps to determine the velocities and directions of objects after collision events, aiding in accident, reconstruction, and vehicle safety design.
• Project Motion: In physics and engineering, the conservation of linear momentum is applied to analyze the motion of projectiles, such as rockets and missiles. Understanding momentum conservation helps to optimize trajectory calculations and predict the behavior of projectiles.
• Fluid Dynamics: In fluid mechanics, conservation of linear momentum is used to analyze the flow of fluids, such as air and water. It also aids in understanding phenomena like drag, lift, and turbulence.
• Aerospace Engineering: Conservation of linear momentum is essential in aerospace engineering for designing spacecraft, satellites, and propulsion systems. It helps engineers calculate the forces and velocities involved in spacecraft maneuvers, orbital transfers, and interplanetary missions, ensuring accurate trajectory planning and control.

Conservation of Linear Momentum Example

Example: Two ice skaters, one with a mass of 80kg and another with a mass of 40kg are initially at rest on a frictionless surface. They push against each other, causing the skater with a mass of 80kg to move to the left with a velocity of -4 m/s. What is the velocity of a skater with a mass of 40 kg?

Solution:

Given that

Before the event

m₁ = 80kg, m₂ = 40kg, v₁ = 0 m/s, v₂ = 0 m/s

∑p_initial = m₁.v₁ + m₂.v₂ = 80×0 + 40×0 = 0 kg.m/s

After the event:

m₁ = 80kg, m₂ = 40kg, v₁ = -4m/s, v₂=? m/s

According to the conservation of linear momentum,

∑p_initial = ∑p_final

0 = 80 × (-4) + 40 × v₂

0 = -320 + 40×v₂

v₂ = 320/40

v₂ = 8 m/s

So, the velocity of a skater with a mass of 40kg after the interaction is 8m/s.

Conservation of Linear Momentum FAQs

What is the principle of conservation of linear momentum?

The conservation of linear momentum states that the total linear momentum of an isolated system remains constant if no external forces act on it.

What are the units of linear momentum?

The SI unit of linear momentum, is kilogram-meter per second (kgm/s).

How is the conservation of linear momentum applied in everyday life?

It is applied in various scenarios such as collisions, rocket propulsion, and the physics of sports to analyze and predict the motion of objects.

How is the conservation of linear momentum related to Newton’s laws of motion?

Conservation of linear momentum is derived from Newton’s laws of motion, particularly the law of conservation of momentum, which states that the total momentum of an isolated system remains constant in the absence of external forces which is based on Newton’s third law of motion

What happens if external forces are present in a system?

If external forces act on a system, such as friction or applied forces, the total momentum of the system may change. In such cases, the change in momentum is equal to the net external force acting on the system, as described by Newton’s second law of motion (F = dp/dt).