Question

Please give me the perfect derivation for 2nd and 3rd law of motion.

Answer 1

Statement-The rate of change of momentum is directly proportional to the force applied on the body and takes place in the direction of the force applied.

Explanation-This law quantifies force it tells us how to measure force in terms of other known quantities which we can measure .

Derivations-Consider a body of mass m moving with initial u .Let a force F acts on the body for time t so that the velocity of the body after time t is v.

Initial momentum of the body pi=mu
Final        "              "   "       "    pf=mv 
Now, change in momentum of the body =p=pf-pi
                                                                    =mv-mu=m(v-u)
Time taken to change this momentum =(t-0)=t
Rate of change of momentum =change of momentum / Time taken 
                                                    = m(v-u)/t
 According to the definition of newton*s second law of motion 
                          Force applied ∞ rate of change of momentum 
or                                      F∞m(v-u)/t
Since                                 v=u+at or (v-u/t)=a
Therefore, eqn. (i) can be written as 
                                        F∞ma
or                                     F=k ma                                                                    ...(ii)
where K is constant of proportionality
If                                    F=1 unit 
                                      m=1 unit
                                       a=1 unit 
Then from eqn.(ii),1=K 
Put this value of K =1 eqm.(ii),we get
                                    F=ma