Question

Prove f= ma where the notation have their usual meaning

Answer 1

Answer -

The formula of force, F = ma is given by Newton's second law of motion.

According to the Newton's second law of motion, rate of change of momentum is directly proportional to the applied unbalanced forces.

$\boxed{\purple{\rm{ Rate\: of\: change \:of \:momentum \propto Force \:Applied }}}$

Forming the equation -

Let a body with initial velocity u and after applying a force F on it its velocity becomes v in time t

$\longrightarrow$Initial momentum of a body = $\sf mu$

$\longrightarrow$Final momentum of body = $\sf mv$

$\longrightarrow$Change in momentum in time = $\sf mv - mu$

Hence, rate of change of momentum

$\sf = \frac{mv - mu}{t}$

But according to Newton's second law,

$\boxed{\sf \frac{mv - mu}{t} \propto F}$

$\implies\sf F \propto \frac{m(v - u)}{t} \: \:\:\:\:\:\:\:\:\:[\frac {v - u}{t} = a]$

$\implies\sf F \propto ma$

$\implies\sf F = kma \:\:\:\:\:\:\:\:\: \\ ( \sf k\: is\: proportionality\: constant)$

$\implies\sf k = 1$

$\boxed{\sf\red{F = ma} }$

Hence proved