Question

mathematical formulation of second law of motion​

Answer 1

To Do :-

• To formulate the second law of motion .

AnsweR :-

$\underline{\color{orange}{\textsf{Newton's \textbf{ Second law of Motion :- }}}}$

This law was given by Sir Issac Newton . According to this law the rate of change of momentum is directly proportional to the applied force in the direction of the force.

If we denote momentum by p , Change In momentum ∆p and Force by F . Then ,

$\sf:\implies F \propto \dfrac{\Delta p}{t}$

Also we know that momentum is product of mass and velocity . Let us take the initial velocity be u and final velocity be v . And mass be m . Overall here our assumptions ,

$\qquad \red{\frak { Let }}\begin{cases} \textsf{ Mass be \textbf{m} .}\\\textsf{ Initial velocity be \textbf{u} .}\\\textsf{ Final velocity be \textbf{v}. } \\\textsf{ Force be \textbf{ F}.}\\\textsf{Force be \textbf{F}.} \end{cases}$

$\purple{\bigstar}\underline{\boldsymbol{ According\ to \ the \ Law \ :- }}$

$\sf:\implies \pink{ Force \propto\dfrac{ \Delta p }{t}} \\\\\sf:\implies F \propto\dfrac{ mv - mu}{t}\\\\\sf:\implies F \propto m \dfrac{ v - u }{t}\qquad \bigg\lgroup \red{\bf a =\dfrac{v-u}{t}}\bigg\rgroup \\\\\sf:\implies F \propto ma \\\\\sf:\implies F = kma \\\\\sf:\implies \boxed{\pink{\frak { Force = mass \times accl^n }}}$