Question

state the newton second laws of motion. derive the mathematical relationship of newton second laws of motion ​

Answer 1

HEY MATE ,

HERE IS YOUR ANSWER,

Newton's Second law of motion :- The rate of change of momentum is directly proportional to the force applied on the system. 

Force applied is directly proportional to the product of mass and acceleration .

Let  be the initial and final momentums respectively.

According to newton's second law :- 

pf - pi / t ∝ F 

We know that, Momentum ( P) = mv .

Let v be the final and u be the initial velocity .

Now, 

mv - mu / t ∝ F 

F ∝ m ( v-u) /t

F ∝ ma. 

F = kma. 

Here, K is the proportionality constant. It's value is 1 .

Units of Force are given by the units of mass and acceleration. Units of force is Kgm/s² .

In accordance to honour the contributions of Newton, 1 kgm/s² is termed as 1 Newton.

Answer 2

Newton's 2nd law of motion states that ;

" The rate of change of momentum is directly proportional to the unbalance force in the direction of force "

$\sf \: Force \propto \dfrac{Change \: in \: momentum}{Time \: taken}$

Consider a body of Mass m having an initial velocity u. The initial momentum of this body will be mu. Suppose a force F acts on this body for time t & causes the final velocity to become v. The final momentum of this body will be mv. Now,the change in momentum of this body is mv - mu & the time taken for this change is t. So, According to Newton's First Law of Motion :

$\large \: \sf \: F \propto \dfrac{ mv \: - \: mu}{t}$

$\implies\large \: \sf \: F \propto \dfrac{ m(v - u)}{t}$

Recall the first equation of motion  

v = u + at

$\implies\tt{a=\dfrac{v-u}{t}}$

Substitute this value in above one

Hence,

$\tt{ F \propto ma }$

But we need to remove the proportionality symbol ,

In order to remove it we need to add an proportionality constant.

So,

$\tt{ F =k* ma }$

k = 1

So,

$\tt{F=m*a}$

Derived.