Question

what is the derivation of Newton's secongld law of motion?​

Answer 1

Solution:

According to second law of the rate of change of momentum is directly proportional to external force applied of an object.

Let Pi is initial momentum & Pf is final momentum, then

$\sf{\implies \dfrac{P_{f}-P_{i}}{t} \propto F_{ex}\;\;\;\;........(1)}$

$\sf{\implies F_{ex}=k\Bigg(\dfrac{P_{f}-P_{i}}{t}\Bigg)}$

$\sf{Here,\; k = 1}$

$\sf{\implies F_{ex}=\dfrac{P_{f}-P_{i}}{t}\;\;\;\;.........(2)}$

$\sf{\implies F_{ex}=\dfrac{\Delta P}{\Delta t}\;\;\;\;.......(3)}$

$\sf{\implies F_{ex}=\dfrac{dp}{dt}\;\;\;\;..........(4)}$

Now, from Equation (2),

$\sf{\implies F_{ex}=\dfrac{mv-mu}{t}}$

$\sf{\implies F_{ex}=m\Bigg(\dfrac{v-u}{t}\Bigg)}$

$\sf{\implies F_{ex}=ma\;\;\;\;.......(5)}$

$\sf{\implies F_{ex}=F}$

Hence Proved!!!!