Momentum Conservation

Introduction

The conservation of momentum is a general law of physics and states that the quantity known as momentum, which characterises motion, never changes in an isolated collection of objects. In other words, a system’s total momentum remains constant. Momentum is equal to the force required to bring an object to stop in a unit length of time and is equal to the mass of an object multiplied by its velocity. 

The total momentum of an array of several objects is the sum of the individual moments. However, because momentum is a vector that includes both the direction and magnitude of motion, the momenta of objects moving in opposite directions can cancel each other, resulting in an overall sum of zero. During the launch, the downward momentum of the expanding exhaust gases equals the magnitude of the rising rocket’s upward momentum, so the system’s total momentum remains constant—in this case, zero. When two particles collide, the sum of their moments before the collision equals their sum after the collision. When one particle loses momentum, the others gain.

Conservation of Linear Momentum

Linear momentum is defined as the product of an object’s mass, “m,” and velocity, “v”. If an object has a lot of momentum, it takes more effort to bring it to a stop. The linear momentum, p, the formula is as follows:

P = m v

The overall momentum is not altered in this case. This is known as momentum conservation.

When no external force is applied to colliding bodies in a given system, the vector summation of specific bodies of linear momentum neither changes nor is affected by their one-to-one interaction, according to the law of conservation of linear momentum.

Formula of Linear Momentum

As previously stated, the particle’s linear momentum is:

p = mv

Furthermore, Newton’s second law for a single particle states:

F = dp/dt ​

Where

F = force of the particle.

The total linear momentum for “n” particles is represented as:

p = p1 + p2 +…..+pn

Individual momentum is denoted by:

m1 v1 + m2v2 + ………..+mnvn

Now, consider the velocity of the centre of mass, which is given by:

V = Σmi​vi​​/Σmi

When we compare the two equations above, we get:

p = mv ——–(i)

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

After differentiating equation i, we get

dp/dt​ = m(dv/dt)​ = ma

where dv/dt is the acceleration of the center of mass, and ma is the external force.

Therefore,

dp/dt = Fext——(ii)

Newton’s second law is represented by Equation (ii) as a system of particles with zero external force acting on it.

So, when Fext = 0 —————(iii)

Then,

dp/dt = 0

The preceding equation demonstrates that p = constant.

As a result, when the force acting on the system is zero, the system’s total linear momentum is either conserved or constant.

Momentum Conversion Formula

The formula of momentum conversion is:

m1u1+m2u2=m1v1+m2v2

The Principle of Momentum Conservation

The law of momentum conservation states that unless an external force is applied, the total momentum of two or more bodies acting on each other in an isolated system remains constant. As a result, momentum cannot be created or destroyed.

Derivation of Momentum Conservation

Newton’s third law states that for every force exerted by an object A on an object B, object B exerts an equal and opposite force in magnitude and direction. Newton used this concept to derive the law of conservation of momentum.

Consider two colliding particles A and B, with masses m1 and m2 and initial and final velocities as u1 and v1 for A and u2 and v2 for B, respectively. The time of contact between the two particles is denoted by ‘t’.

A = m1(v1−u1)(change in momentum of particle A)

B =m2(v2−u2) (change in momentum of particle B)

FBA= −FAB (from Newton’s third law of motion)

FBA=m2a2=m2(v2-u2)/t , FAB=m1a1=m1(v1-u1)/t

m1u1+m2u2=m1v1+m2v2

As a result, the equation of the law of conservation of momentum is as follows: m1u1 + m2u2 represents the total momentum of particles A and B before the collision, and m1v1 + m2v2 represents the total momentum of particles A and B after the collision.

Examples of the Law of Conservation of Momentum

The following are examples of the law of conservation of momentum:

1. Rocket propulsion: A gas chamber is located at one end of a rocket. This chamber ejects gas at breakneck speed. Before the gas is ejected, the total momentum is zero. The rocket gains recoil velocity and acceleration in the opposite direction due to the ejection of gas from the rocket. This is due to the momentum conservation principle.

1. Balloon: The small particles of gas move quickly, colliding with one another and the balloon’s walls. Regardless of whether the particles themselves move faster or slower when they lose or gain momentum when they collide. The system’s overall momentum remains unchanged. As a result, even if we add outside energy to the balloon by heating it, its size does not change. The balloon should expand because it increases the speed of the particles, which expands their force and thus increases the force applied to the balloon’s walls.

1. Recoil of a gun: When a bullet is fired from a gun, the bullet and gun are initially still, implying that the total momentum prior to firing is zero. When the bullet is discharged, it gains forward momentum. According to the law of conservation of momentum, the gun gains regressive momentum. The bullet has a mass of m and a velocity of v. The mass M gun achieves a retrogressive backward velocity v. The total momentum is zero prior to the termination. As a result, the total momentum after firing is zero as well.

0 = mv+Mu

u= −(m/M) v

Essentially, u is the velocity of recoil. The mass of the bullet is smaller in comparison to the mass of the gun, m M. The gun’s backward velocity, u, is extremely low.

Conclusion

Momentum conservation is a fundamental physical law. It denotes that the total momentum of a disconnected or isolated system/framework is conserved. As a result, if no external force acts on the system, the total momentum of the system remains constant during an interaction. A system’s total momentum is the vector sum of its momenta. As a result, the total momentum component in any direction remains constant. In all physical processes, momentum is conserved. The law of conservation of momentum has been amply confirmed by experiment. It can even be mathematically deduced on the reasonable assumption that space is uniform—that nothing in nature’s laws distinguishes one position in space from another.