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 )