derive the formula for force which …

According to Newton's Second Law: If a particle of mass {tex}m{/tex} has velocity {tex}v{/tex} at any moment {tex}t{/tex},

{tex}\vec{F}=\frac{d\vec{p}}{dt}{/tex}, where {tex}\vec{F}{/tex}is the force and {tex}\vec{p}=m\vec{v}{/tex} is the momentum.

Thus,

{tex}\vec{F} =\frac{d(m{\vec{v}})}{dt} =m\frac{d\vec{v}}{dt}+\vec{v}\frac{dm}{dt}{/tex}

For cases where mass changes with time, the above equation would be applicable.

Special Cases:

• If mass remains constant, as we see in most theories an numericals, {tex}\frac{dm}{dt}=0{/tex} hence, {tex}\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}{/tex}, where {tex}\vec{a}{/tex} is the acceleration of the object.
• If the velocity remains constant but mass emerges out or gets added at a constant rate, as in cases of water coming out of hose pipe, or sands added at constant rate upon a moving belt; {tex}\vec{F}=\vec{v}\frac{dm}{dt}{/tex}.