Conservation of Momentum for JEE

In Classical Physics, if there is a moving body and it possesses some mass, it will hold some momentum. In simple terms, momentum gives an idea of the force needed to stop the object or to keep it in motion. Every moving body carries the momentum of its own. Momentum is a vector quantity as it is the product of the mass and velocity of a body. If no external force is applied to the body, the momentum will remain unchanged and will be conserved for that very system of masses.

Law of Conservation of Momentum

The law of conservation of momentum states that for an isolated system of particles, the total momentum will remain the same. Any internal collisions or movements will not affect the momentum in any way. The momentum of the system will be an unchanging value unless an external force is applied.

We can approach to prove the statement for our sake. Imagine two objects colliding, namely $A$ and $B$. For the collision, considering that Newton's third law of motion holds true. The forces acting between two objects will be equal in magnitude and opposite in direction (Action–Reaction pair). Let’s call the net force exerted on Object A during a collision is ${{F}_{1}}$ and for Object $B$ is ${{F}_{2}}$.

We can safely say ${{F}_{1}}=\text{ }-\text{ }{{F}_{2}}$

Moving further, the forces that are acting on two objects will only act for the time collision is happening. This time length is not our concern, our only concern is that it is going to be equal for both the objects.

${{t}_{1}}=\text{ }{{t}_{2}}$

So, the total impulse transferred to object A shall be ${{F}_{1}}{{t}_{1}}$ and for object B be ${{F}_{2}}{{t}_{2}}$.

The net impulse transferred to the system remains zero as  

${{F}_{1}}=\text{ }-\text{ }{{F}_{2}}$  so ${{F}_{1}}{{t}_{1}}=\text{ }-\text{ }{{F}_{2}}{{t}_{2}}$

We can write the following as ${{m}_{1}}\Delta {{v}_{1}}=-{{m}_{2}}\Delta {{v}_{2}}$

The momentum changes are equal and in opposite direction; hence, the momentum

of the system remains conserved.

For an isolated system of masses, if any internal forces act or there is a change in the individual momenta of masses, the net change in momentum will be zero.

For an isolated system with n number of masses,

${{m}_{1}}\Delta {{v}_{1}}+{{m}_{2}}\Delta {{v}_{2}}+{{m}_{3}}\Delta {{v}_{3}}..........+{{m}_{n}}\Delta {{v}_{n}}=0$

$\sum\limits_{i=1}^{n}{{{m}_{i}}\Delta {{v}_{i}}}=0$

This is known as the law of conservation of linear momentum.

Explosion of a Bomb

A bomb is a system of masses itself. While exploding, the whole mass breaks into smaller masses and scatters with their individual velocities. The explosion takes place in a bomb and the energy from the explosion is converted into the kinetic energy of the scattered individual particles. If a bomb is at rest before an explosion, then it will be having zero momentum. So, after exploding, the whole system must have zero momentum because all the forces generated during the explosion are internal and cancel out each other, so the law of conservation of linear momentum will hold true for the case. 

Just after exploding, the bomb will scatter into fragments and each fragment will have its own momentum vector. These momentum vectors will be scattered all across the surrounding space in arbitrary directions. But as the total momentum is conserved and zero, all the individual momentum vectors must add to zero. If M is the total mass of the bomb,

$\sum\limits_{i=1}^{n}{{{m}_{i}}=M}$and ${{m}_{1}},\text{ }{{m}_{2}}\ldots \ldots \ldots {{m}_{n}}$ is the mass of the individual fragments.

And, 

$\sum\limits_{i=1}^{n}{{{m}_{i}}{{\overset{\to }{\mathop{v}}\,}_{i}}=0}$ (Vector sum of momentum vectors of individual fragments)

Hence, the total linear momentum will be conserved.

Examples of Conservation of Momentum

• A rocket ejects burning gases at high speed to accelerate. The reason behind this is the gases have mass and while ejecting these gases are at very high velocity and carry a high momentum. So, to conserve momentum, the speed of the rocket increases in the direction opposite to the ejection. 

• While firing a bullet from a gun, the shooter feels a recoil. It is because, for a gun as a system, the total momentum of the gun remains the same. So, when a bullet leaves the gun at high velocity, the equal and opposite impulse is imparted to the gun and it recoils.

• A firefighter is always advised to hold the pipe tight. It is because the water leaves the nozzle at a very high speed, so the pipe experiences a constant backward impulse in order to conserve the momentum.

Application of Conservation of Momentum

• One of the main applications of conservation of momentum is while launching rockets. The rocket fuel is burned and exhaust gases are pushed downwards and due to this, the rocket is pushed upwards. When the exhaust gases are ejected at high velocity, an equal and opposite impulse is transferred to the rocket.

• Motorboats work on the same principle of conservation of momentum. The motor pushes the water backwards and as a result, the boat moves forward. Pushing water backwards imparts equal and opposite impulse to the motor boat and the boat accelerates.

• In aeroplanes, engines push the exhaust gases backwards which results in moving the aeroplane forward. It is the same as other cases, leaving gases impart a forward impulse to the plane.

Conclusion

The law of conservation of momentum is one of the important and fundamental laws of Physics. It is observed in many real-life situations. From the explosion of lethal bombs to small firecrackers, the conservation of momentum holds true. It is an advantage in many real-life situations. The momentum conservation equations are the base of many important mathematical models like fluid dynamics, nuclear physics, and so on. Thus, it holds a great importance and a great advantage in the realm of Physics.