## Linear momentum

We have seen quantities like velocity, acceleration, mass etc. and know what they signify. Now we will define a quantity known as momentum which is mathematically represented as:

**P = mv**

which is the amount of motion a body possess. As it is a product of mass and velocity so like velocity it is also a vector quantity. Its unit is Kg m s^{-1}. Now we will relate the kinetic energy of a body with its linear momentum.

**P = mv**

Squaring both sides and dividing by 2 we get,

So if two bodies having different masses have the same kinetic energy than the one with lighter mass has smaller momentum.

## How to calculate the Rate of Change of Momentum?

Newton’s Second law relates force with the rate of change of momentum. According to the law, force is directly proportional to the rate of change in momentum.

We will use this to state law of conservation of momentum. According to this if the net force acting on the system is zero then the momentum of the system remains conserved. In other words, the change in momentum of the system is zero. We can see as F = 0 so will also be zero according to the second law. Let’s take the following example:

We consider m_{1} and m_{2} as our system. So during the collision, the net force on the system is zero and hence we can conserve momentum of the system. The equation for momentum will be:

*Initial momentum* = m_{1}u_{1} + m_{2}u_{2}

*Final momentum* = m_{1}v_{1} + m_{2}v_{2}

*So according to the conservation of momentum,*

m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}

But one thing to take care is that conservation is only true for a system and not one body because if we consider only a single body m_{1} or m_{2}, then net force will be acting on it so we cannot write

m_{1}u_{1} ≠ m_{1}v_{1} or m_{2}u_{2} ≠ m_{2}v_{2}.

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