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Conservation of Momentum

One of the most important and powerful laws in physics is the law conservation of momentum. This law can be stated as follows. For a collision occurring between an object denoted as O1 and another object denoted as O2 in an isolated system, total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object O1 is equal to the momentum gained by object O 2. From the above statement we can conclude that the total momentum of a collection of objects or a system is conserved that means the total amount of momentum is an unchanging value or a constant. So now let’s start with the logical explanation of conservation of momentum. Let us assume a collision between two objects O1 and O2. For such a collision, according to Newton’s 3rd law the forces acting between the two objects are equal in magnitude and opposite in direction .This statement can be written as follows: F₁=-F₂

Now consider that the forces act between the two objects for a given amount of time. This time may be long or short depends upon different cases. But regardless of amount of time it can be said that the time that the force acts upon object O1 is equal to the time that the force acts upon object O2. As equation this can be written as: t₁=t₂

The forces acting between the two objects O1 and O2 are equal in magnitude but opposite in direction, as the times for which these forces act are equal in magnitude, it can be said that impulses experienced by the two objects are also equal in magnitude and opposite in direction. This can be written as: F₁*t₁=-F₂*t₂

But we know that the impulse experienced by an object is equal to the change in momentum of that object. Since each object experiences equal and opposite impulses, logically they must also experience equal and opposite momentum changes. As an equation, this can be stated as: m₁*∆v₁=-m₂*∆v₂, where v₁ and v₂ are the velocities of objects O1 and O2 respectively. In general for collision of two particles, by using momentum conservation law, it can be said that

m₁ u₁+m₂ u₂=m₁ v₁+m₂v₂

where u₁ and u₂ are the velocities of objects O1 and O2 before collision, and v1 and v2 are the velocities after collision.

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