Conservation of Linear Momentum

Conservation of Linear Momentum

          Momentum

    The product of mass and velocity of a moving body is said to be momentum. 

It is generally denoted by P. 

The SI unit of momentum is kilogram metre per second or kgms⁻¹. 

The dimension of momentum is [MLT⁻¹]

          Law of conservation of Linear Momentum

"The vector sum of the linear momenta of all the particles in an isolated system remains constant in the absence of any external force"

          Proof:

    Consider about an isolated system having n particles. Let the masses of the particles be M1, M2, M3 ..... Mn and their velocities V1, V2, V3 ....... Vn respectively. 

The vector sum of the linear momenta of all the particles in the system is given by,

P  = M1V1 + M2V2 + M3V3 + ....................... + MnVn .......................... (i)

Let the total mass of the system is M and velocity of the centre of mass of the system is Vcm

Form of Equation (i) will be

P = MVcm 

Now derivative both sides with respect to t

dP/dt = M(Vcm/dt)

dP/dt = Macm (acm is the acceleration of centre of mass of system)

dP/dt = Fext (Fext is external force applied on the system)

When Fext is Zero

dP/dt = 0

We know that derivative of a constant term is zero.

Therefore,

P = Constant

Hence,

P =  M1V1 + M2V2 + M3V3 + ....................... + MnVn  = Constant.