Newton's laws of motion

This section is devoted to derivation of newton's laws of motion by calculus method,which will involve little use of differential and integral calculus.

Derivation 1. We know that v=dx/dt

Where v is the velocity and dx is small change in displacement with respect to the small change in time.

We have

V=lim x=0 Δx/Δt=dx/dt

By definition of limit and derivative.

Now,v=dx/dt

V.dt=dx

We know that v=u + at,so we have

(u + at) dt=dx

U.dt + at.dt=dx

Now by integrating both sides, we have

Integrating from 0 to x dx=integrating from 0 to t udt + integrating from 0 to t at.dt

X=ut +1/2at.t

As we know by standard integral formula's.

Now derivarion 2.

We know  a=dv/dt

By multiplying and dividing R.H.S. by Dx,we have

a=dv/dt.dx/dx

a=dv/dx.dx/dt

a=v.dv/dx

a.dx=v.dv

Again by integrating

Integrating from 0 to x w.r.t the dx a.dx=integrating from u to v v.dv

a.x=v.v-u.u/2

2.a.x=v.v-u.u

Which is another equation of motion.

Third derivation

Simply a=dv.dt

a.dt=dv

Now integrating within limits

Integrating from 0 to t a.dt=integrating from u to v dv

at=v-u

v=u + at

By standard formula use.

These are three derivations by calculus method,graphical method as well as analytical method can also be used to obtain the same result over and over again.