Linear Momentum

Linear momentum is defined as the product of mass of the particle and its velocity, it is a vector quantity.
 \overrightarrow{p} = m \overrightarrow{v} 
From newton's second law of motion, for a fixed mass particle
 \overrightarrow{F} = m \overrightarrow{a} = m \frac{d \overrightarrow{v}}{dt} = \frac{d}{dt} \left( m \overrightarrow{v} \right) 
For a system of n particles with masses m_1, m_2, m_3, ..., m_n and velocities \overrightarrow{v_1}, \overrightarrow{v_2}, \overrightarrow{v_3}, ..., \overrightarrow{v_n} respectively, the the net momentum of the system is:-
 \overrightarrow{p_{net}} = m_1 \overrightarrow{v_1} + m_2 \overrightarrow{v_2} + m_3 \overrightarrow{v_3} + ... + m_n \overrightarrow{v_n} = \overrightarrow{p_1} + \overrightarrow{p_2} + \overrightarrow{p_3} + ... + \overrightarrow{p_n}
 \overrightarrow{p_{net}} = M \overrightarrow{V_{cm}} 
Differentiating the above expression with respect to time:
 \frac{d}{dt} \left( \overrightarrow{p_{net}} \right) = M \frac{d}{dt} \left( \overrightarrow{V_{cm}} \right) \overrightarrow{F_{net}} = M \overrightarrow{a_{cm}}
and also
 \overrightarrow{F_{net}} = \frac{d}{dt} \left( \overrightarrow{p_{net}} \right) 
The magnetic of linear momentum may be expressed in terms of kinetic energy as well:-
==Law of Conservation of Momentum==
If the net force acting on a body is equal to zero, then the momentum of the body remains constant. This is known as law of conservation of momentum.
 F_{net} = 0 \frac{d}{dt} \left( p_{net} \right) = 0 
therefore, p = 0 or p = constant.
If the velocity of centre of mass is constant (v = constant) then we get \overrightarrow{p_{net}} = constant.
This is known as the law of conservation of linear momentum of the system of particles.
Category:Mechanics