Newton’s 2nd law:
The rate of change of momentum is directly proportional to the impressed force and is in the same direction along which the force acts.
The magnitude is referred in the first part of the law, and the direction is referred in the second part of the law.
Let us consider a particle of mass \( m \) is moving with velocity \( \vec{v} \) due to the application of an external force \( \vec{F} \)
Then we can write according to the 2nd law,
\( \displaystyle{\vec{F}=\frac{d}{dt}(m\vec{v})=\frac{d\vec{p}}{dt}} \)
where, \( \vec{p}=m\vec{v} \) is the linear momentum of the particle.
If \( m \) is independent of time \( t \) then we can write,
\( \displaystyle{\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}} \)
where, \( \displaystyle{\vec{a}=\frac{d\vec{v}}{dt}} \) is the acceleration of the particle.
From the time of Sir Newton to early in the twentieth century, the mass of the body was considered as constant and independent of its velocity. But in 1950, Sir Einstein, with the help of relativity, showed that the mass is not constant, it is relative to the observer and varies with its velocity.
In this case, Newton’s 2nd law can be written as,
\( \displaystyle{\vec{F}=\frac{d}{dt}(m\vec{v})} \)
\( or,\displaystyle{\vec{F}=m\frac{d\vec{v}}{dt}+\vec{v}\frac{dm}{dt}} \)
But in practical world, the velocity of most of the bodies is extremely small as compared to the velocity of light, so the change in mass is negligibly small.