Question

Prove F = m x a mathematically​

Answer 1

According to Netwon's Second Law of Motion, The rate of change of momentum of body is directly proportional to the applied force and it takes place in the direction in which the force acts.

$\bf \implies \overrightarrow{F} \propto \dfrac{d \overrightarrow{p}}{d t} \\ \\ \rm \implies \overrightarrow{F} = k \dfrac{d \overrightarrow{p}}{dt} \\ \\ \rm \implies \overrightarrow{F} = k \dfrac{d (m \overrightarrow{v})}{dt} \\ \\ \rm \implies \overrightarrow{F} = k m\dfrac{d \overrightarrow{v}}{dt} \\ \\ \rm \overrightarrow{a} = \dfrac{d \overrightarrow{v}}{dt} : \\ \\ \rm \implies \overrightarrow{F} = km \overrightarrow{a}$

The value of constant of proportionality k is considered as 1. So, By taking k = 1 we get:

$\boxed{ \bf \overrightarrow{F} = m \overrightarrow{a}}$