Verbal proof of "Law of conservation of momentum".

Hmmm. The intension here is to demonsrate the law of conservation of momentum more intutively. Newton's 3 laws of motion can be understood quite intutively. When a force is applied, a body changes its motion. If there was no force, its motion wouldn't have changed. It can be imagined quite easily. But, when we go one step ahead, i.e. to Law of Conservation of Momentum, things tend to become more mathematical. And One makes a statement "the velocity of a body should be blah blah m/s because the momentum of the system should get conserved." I am sure, in many cases the student who makes this statement doesn't understand much from it. The student is perfect mathematically. But does he really understand the physics behind it?

In this little blog, I will try to expalin the proof of Law of conservation of momentum verbally and so more intutively. The raw material for the proof is going to be only Newton's 3 laws of motion! So, before starting with the actual proof, I will state the 3 laws.

1st law: Every body continues to be in its state of rest or of uniform motion along a straight line unless it is acted upon by an external unbalanced force.
This law defines force qualitatively.

2nd law: The rate of change of momentum is directly proportional to force applied and the change takes place in the direction of force.
This law defines force quantitatively.

3rd law: For every action there is an equal and opposite reaction.

Momentum = mass * velocity

Law of conservation of momentum: In a system of bodies, the total momentum remains constant if no external unbalanced force acts on the system.



Proof starts ------X--------X--------

To start with, consider one single body. What happens if no force acts on it? Well, 1st law states that its velocity should remain constant. And its mass is also constant (We won't consider relativity). So, mass times velocity that is the momentum remains constant.

But, what if there are a number of bodies in the system? They can affect each others' motions through collisions or by gravitational / electromagnetic forces. In this case, 3rd law comes to our rescue! According to it, the forces in a pair of interacting bodies are equal and opposite. We have extracted this important information that, 2 bodies - whatever may be their masses - apply equal and opposite forces on each other. The 2nd law now does rest of the work. According to it, rate of change of momentum of a body is directly proportional to force applied on it. So for a given amount of time "Change in momentum of a body is directly proportional to the force applied". In case of 2 bodies interacting, they apply force on each other for equal amounts of time. So, both the bodies experience equal and opposite forces for equal amounts of times. As a result, they are subjected to equal change in momentum in opposite directions. That is, if momentum of body A increases by some amount then for body B it must reduce by same amount. In other words, velocities of bodies change in inverse proportion of their masses. So, product of mass and velocity remains constant. As a result, the total momentum gets conserved.

What if there are more than 2 bodies? Principle of superposition applies here. That is net effect on a body is combined effect because of all bodies considered individually. Above 3 paragraphs apply to "all pairs" in the system. Superimposing them, the total momentum gets conserved.

Proof ends ------X--------X---------

Law of conservation of momentum is really a very handy law. It gives a simple solution to complex system. Working out individual interacting particles by analysing forces between them would become very tidious. Instead we can consider the whole system of particles at one go and work out the solution. But, in doing so there is a chance that the student does it totally mathematically and moves away from the underlying science behind it. I hope this blog clears the concept.

Ameya L. Gokhale