Statement : Time rate of change of momentum of a body is proportional to resultant force acting on it this change is in direction of resultant force.

Let resultant force $\mathrm{F}$ act on body of mass $m$ for $\Delta t$ time interval. During this let its velocity change from $\vec{v}$ to $\underset{\rightarrow}{\vec{v}}+\Delta \vec{v}$.

Initial momentum $\vec{p}_{i}=m \vec{v}$

Final momentum $p_{f}=m(\vec{v}+\Delta \vec{v})$

$\therefore$ Change in momentum

$\Delta p =p_{f}-\vec{p}_{i}$

$=m(v+\Delta v)-m v$

$=m v+m \Delta v-m v$

$=m \Delta v$

From second law of motion,

$\overrightarrow{\mathrm{F}} \propto \frac{\Delta \vec{p}}{\Delta t}$

$\overrightarrow{\mathrm{F}}=k \frac{\Delta \vec{p}}{\Delta t}$ where $k$ is constant of proportionality

By taking $\lim _{\Delta t \rightarrow 0}$

$\overrightarrow{\mathrm{F}} =\lim _{\Delta t \rightarrow 0} k \frac{\Delta \vec{p}}{\Delta t}$

$\overrightarrow{\mathrm{F}} =k \frac{d \vec{p}}{d t}$

$\therefore \overrightarrow{\mathrm{F}} =k \frac{d(m \vec{v})}{d t}$

$=k m \frac{d \vec{v}}{d t}+k \vec{v} \frac{d m}{d t}$

$=k m \frac{d \vec{v}}{d t}[\text { Considering mass as constant }]$

$\therefore  \overrightarrow{\mathrm{F}}=k m \vec{a}$

$\therefore  \overrightarrow{\mathrm{F}} \propto m \overrightarrow{\vec{a}}$

Thus, force acting on a body is proportional to product of mass and velocity.

$(i)$ Newton's second law of motion,

$\overrightarrow{\mathrm{F}}=\frac{d \vec{p}}{d t}$

If mass is considered constant,

$\overrightarrow{\mathrm{F}}=m \vec{a}=m\left(\frac{\overrightarrow{v_{2}}-\overrightarrow{v_{1}}}{\Delta t}\right)$

$(ii)$ If resultant external force acting on body is zero then in,

$\overrightarrow{\mathrm{F}}=m \vec{a}$

$\overrightarrow{\mathrm{F}}=0$

$\vec{a}=0$

v=\text { constant }

which is consistence with Newton's first law of motion.

$(iii)$ Newton's second law of motion gives magnitude of force. Force is vector quantity. Component of force are :

X-comp. $\mathrm{F}_{x}=\frac{d p_{x}}{d t}=m a_{x}$

Y-comp. $\mathrm{F}_{y}=\frac{d p_{y}}{d t}=m a_{y}$

Z-comp. $\mathrm{F}_{z}=\frac{d p_{z}}{d t}=m a_{z}$

This means that when force is acting at by making angle with initial velocity, then component in direction of initial velocity will change but perpendicular component will remain constant.

For example, in case of projectile motion gravitational force act in downward direction and horizontal component remain constant.

$(iv)$ Equation $\mathrm{F}=\frac{d p}{d t}=m a$ applied to point like object where $\mathrm{F}$ is resultant force acting on particle and $a$ is acceleration produced.

– This law can be applied to rigid body or system of particles also where $\mathrm{F}$ is resultant force on system $a$ is acceleration of system of particles.

$(v)$ Newton's second law of motion is for object (body) at given place force $(F)$ has relation with acceleration $(a)$ at given instant. Thus, acceleration can be obtained from force, but it cannot be determined from history of motion of object.