# 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 â€˜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 systemâ€™.

This law can be deduced from Newtonâ€™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) Newtonâ€™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 m_{1} and m_{2} moving with velocities v_{1} and v_{2} 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 m_{1} is p_{1} = m_{1}v_{1 }

Linear momentum of mass m_{2} is p_{2} = m_{2}v_{2 }

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

or p = m_{1}v_{1} + m_{2}v_{2 }

Differentiating with respect to time, we get

where **F _{1}**and

**F**are the forces acting on m

_{2}_{1}and m

_{2}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 Newtonâ€™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 Newtonâ€™s second law, these external forces will change the momentum of the particles. Thus, we conclude that forces **F _{1}**and

**F**in equation (3.3) are actually the external forces acting on the particles.

_{2}The resultant external force **F**_{ext} is given by the vector sum

F_{ext} = **F**_{1} + **F**_{2 }

Hence, eq. (3.3) becomes

If **F**_{ext} = 0, we have

or **p** = constant

This implies that the vector sum **(p = p _{1} + p_{2}) **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 m_{1,}m_{2} â€¦, m_{n }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 v_{1}, v_{2}, â€¦, v_{n} respectively. Their total momentum p is equal to the vector sum of their individual momenta **p**_{1,}**p**_{2}, â€¦, **p**_{n}, i.e.

**P** = **p**_{1} +** p**_{2} +â€¦+ **p**_{n }

P = m_{1}v_{1} + m_{2}v_{2} +....+ m_{n}v_{n} (Q p = mv)

Differentiating with respect to t we have

_{}

where

**a**

_{1},

**a**

_{2}, â€¦,

**a**

_{n }are the respective accelerations.

Now from Newtonâ€™s second law, **F**=m**a**.

Thus

where **F**_{1}, **F**_{2}, and **F**_{n} 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.

**F**_{1}, **F**_{2}, and **F**_{n} are only the external forces acting on the particles. The net external force **F**_{ext} is clearly equal to the vector sum of all these forces, i.e.

F_{ext} = **F**_{1 }+ **F**_{2 }+ ... + **F**_{n }

Thus

If **F**_{ext} = 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 Newtonâ€™s second law of motion. We shall now show that Newtonâ€™s third law is a consequence of the law of conservation of linear momentum.

# Deduction of Newtonâ€™s Third Law of Motion from the Law of Conservation of Momentum

Consider an isolated system of two bodies of masses m_{1} and m_{2} mutually interacting with each other. These interactions cause a change in the velocities and hence a change in the momentum of each body. Let Î”**p _{1}** and Î”

**p**denote the change in momentum of m

_{2}_{1}and m

_{2}respectively, brought about in a time interval Î” 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.

Î”**p _{1}** + Î”

**p**= 0

_{2}or Î”**p _{2}** = -Î”

**p**

_{1}

_{ }Dividing by Î”t and taking the limit as Î”t Î” 0, we have

or

i.e. rate of change of momentum of m_{2 }= - rate of change of momentum m_{1 }

or Force on m_{2} = - Force of m_{1 }

or Action = - Reaction

i.e. the action and reaction are always equal and opposite. This is Newtonâ€™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 m_{b }and m_{g} are respectively the masses of the bullet and the gun and if v_{b} and v_{g} are their respective velocities, then the total momentum of the bullet-gun system after firing is ** m _{b}v_{b }+ m_{g}v_{g}**.

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

m_{b}v_{b }+ m_{g}b_{g }= 0

It is clear from this equation that v_{g} is directed opposite to v_{b.} Knowing m_{b}, m_{g} and v_{b} the recoil velocity v_{g} 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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