1. Derivation of First Equation of Motion by Calculus Method

Since acceleration is the rate of change of velocity, it can be mathematically written as:

a=dvdta=dvdt

Rearranging the above equation, we get

adt=dvadt=dv

Integrating both the sides, we get

∫t0adt=∫vudv∫0tadt=∫uvdv

at=v−uat=v−u

Rearranging, we get

v=u+atv=u+at

2. Derivation of Second Equation of Motion by Calculus Method

Velocity is the rate of change of displacement.

Mathematically, this is expressed as

v=dsdtv=dsdt

Rearranging the equation, we get

ds=vdtds=vdt

Substituting the first equation of motion in the above equation, we get

ds=(u+at)dtds=(u+at)dt

ds=(u+at)dt=(udt+atdt)ds=(u+at)dt=(udt+atdt)

On further simplification, the equation becomes:

∫s0ds=∫t0udt+∫t0atdt∫0sds=∫0tudt+∫0tatdt

=ut+12at2

3. Derivation of Third Equation of Motion by Calculus Method

a=dvdta=dvdt (1)

We also know that velocity is the rate of change of displacement and can be represented as:

v=dsdtv=dsdt (2)

Cross multiplying (1) and (2), we get

adsdt=vdvdtadsdt=vdvdt

∫s0ads=∫vuvds∫0sads=∫uvvds

as=v2−u22as=v2−u22

v2=u2+2as

Answer Expert : Dwayne Smith ( Statistics Assignment Help Expert @LivewebTutors )