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Conservation of Momentum

The law of conservation of momentum is the most fundamental conservation law. Newton took the trouble of defining a new quantity called the momentum p as the product of mass m and velocity v, because, this particular combination is useful for understanding the laws of nature. What makes momentum so important is the fact that it is conserved (i.e. it remains constant).

The law of conservation of linear momentum may be stated as ‘when no net external force acts on a system consisting of several particles, the total linear momentum of the system is conserved, the total linear momentum being the vector sum of the linear momentum of each particle in the system’.

This law can be deduced from Newton’s laws of motion when generalized to a system of more than one particle (or body).

One Body System

For a system consisting of a single particle, Eq. (3.2) Newton’s second law can be rewritten as

F = ma

or

where p = mv is the linear momentum of the particle. Now if the external force F applied to the particle is zero, it is clear that

or p=constant, showing that in the absence of an external force, the linear momentum of a particle remains constant.

Two-body System

Consider a system of two particles of masses m1 and m2 moving with velocities v1 and v2 respectively in the same straight line. These particles may collide with each other, and as a result their velocities may change. At any instant of time, we have

Linear momentum of mass m1 is p1 = m1v1

Linear momentum of mass m2 is p2 = m2v2

Thus the total linear momentum (p) = vector sum of the individual linear momentum of each particle

or p = m1v1 + m2v2  

Differentiating with respect to time, we get

where F1and F2are the forces acting on m1 and m2 respectively.

In a system consisting of two particles, there can be two kinds of forces: (i) internal force and (ii) external forces.

Internal forces are the forces that the particles exert on each other during their interactions (e.g. collisions). From Newton’s third law, these forces always occur in pairs of action and reaction. Since these forces are equal and opposite, they bring about equal and opposite changes in the momentum of the particles. Thus, internal forces cannot bring about any net change in the momentum of a particle.

The external forces, on the other hand, are the forces exerted from outside the system of two particles. From Newton’s second law, these external forces will change the momentum of the particles. Thus, we conclude that forces F1and F2in equation (3.3) are actually the external forces acting on the particles.

The resultant external force Fext is given by the vector sum

Fext = F1 + F2

Hence, eq. (3.3) becomes

If Fext = 0, we have

or p = constant

This implies that the vector sum (p = p1 + p2) of the linear momentum of the particles remains constant, if the net external force is zero.

Many-body System

The law can be extended to any number of particles. These particles may be the particles of a rigid body in which their positions are fixed or they may form an assembly and be free to move freely in any direction.

Let m1,m2 …, mn be the masses of the different particles of the body and, in addition to internal forces due to their own interactions with each other, let there also be an unbalanced external force acting on them.

As a result of this external net force, they acquire velocities v1, v2, …, vn respectively. Their total momentum p is equal to the vector sum of their individual momenta p1,p2, …, pn, i.e.

P = p1 + p2 +…+ pn

P = m1v1 + m2v2 +....+ mnvn (Q p = mv)

Differentiating with respect to t we have

 

where a1, a2, …, an are the respective accelerations. 

Now from Newton’s second law, F=ma.

Thus

 

where F1, F2, and Fn are the forces acting on individual particles. These forces include both internal and external forces. The internal forces, which arise due to mutual inter-particle interactions, always occur in pairs.
 

Since these action and reaction forces are equal and opposite, they bring about equal and opposite changes in the momentum of the particles of the system. Thus, the internal forces cannot change the momentum of the particles.
 

F1, F2, and Fn are only the external forces acting on the particles. The net external force Fext is clearly equal to the vector sum of all these forces, i.e.

Fext = F1 + F2 + ... + Fn

Thus

 

If Fext = 0, it follows that = 0 or p = constant, i.e. the total linear momentum (which is equal to the vector sum of the momentum of individual particles) of the system is constant, if the net external force acting on the system is zero. This is the law of conservation of linear momentum.
 

A system which is free from the influence of any external forces is called an isolated system.

We have seen that the law of conservation of linear momentum can be derived from Newton’s second law of motion. We shall now show that Newton’s third law is a consequence of the law of conservation of linear momentum.

Deduction of Newton’s Third Law of Motion from the Law of Conservation of Momentum

Consider an isolated system of two bodies of masses m1 and m2 mutually interacting with each other. These interactions cause a change in the velocities and hence a change in the momentum of each body. Let Δp1 and Δp2denote the change in momentum of m1 and m2 respectively, brought about in a time interval Δ t.
 

Now, according to the law of conservation of linear momentum, the net change in the linear momentum in an isolated system must be zero, i.e.

Δp1 + Δp2 = 0

or Δp2 = -Δp1

 

Dividing by Δt and taking the limit as Δt Δ 0, we have

or

 

i.e. rate of change of momentum of m2 = - rate of change of momentum m1

or Force on m2 = - Force of m1

or Action = - Reaction

 

i.e. the action and reaction are always equal and opposite. This is Newton’s third law of motion

Some Illustrations of Conservation of Momentum

The following examples illustrate the law of conservation of momentum.


Recoil of a Gun: The gun and the bullet constitute a single system. Before the gun is fired, both the gun and the bullet are at rest. Therefore, their total momentum is zero. When the gun is fired, the bullet moves forward and the gun recoils backwards. 


If mb and mg are respectively the masses of the bullet and the gun and if vb and vg are their respective velocities, then the total momentum of the bullet-gun system after firing is mbvb + mgvg


According to the law of conservation of momentum, the total momentum before firing must be equal to that after firing, i.e.

mbvb + mgbg = 0

It is clear from this equation that vg is directed opposite to vb. Knowing mb, mg and vb the recoil velocity vg of the gun can be determined.

Rockets


The propulsion of rockets and jet planes is an interesting application of the conservation law. When the fuel of the rocket is exploded, gases escape with a large velocity and hence a large momentum. The escaping gases, in turn, impart an equal and opposite momentum to the rocket.

Explosion of a bomb


Suppose a bomb is at rest before it explodes. Its total momentum is zero. When it explodes, it breaks up into many parts, each part having a particular momentum. For a part flying in one direction with a certain momentum, there is another part moving in the opposite direction with the same momentum. If the bomb explodes into two equal parts, they will fly off in exactly opposite directions with the same speed (since each part has the same mass).





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