Derive equations of uniformly acceleration motion by calculus method.

Explanation

Consider an object moving in a straight line with uniform or constant acceleration 'a'. Let u be the velocity of the object at time t=0, and v be velocity of the body at a later time t.
Velocity - time relation
(i) The acceleration of the body at any instant is given by the first derivative of the velocity with respect to time,
`a= (dv)/(dt)" or "dv=a.dt`
Integrating both sides with the condition that as time changes from 0 to t, the velocity changes from u to v. For the constant acceleration,
`int_(u)^(v) dv= int_(0)^(t) a dt = a int_(u)^(v) dt implies [v]_(u)^(v) = a [t]_(0)^(t)" ""............."(1)`
`v-u = at" or " v= u+at`
Displacement - time relation
(ii) The velocity of the body is given by the first derivative of the displacement with respect to time.
`v= (ds)/(dt)" or "ds = vdt` and since `v=u+at`
We get `ds = (u+at)dt`
Assume that initially at time t=0, the particle started from the origin. At a later time t, the particle displacement is s. Further assuming that acceleration is time - independent, we have
`int_(0)^(s) ds= int_(0)^(t) u dt +int_(0)^(t) at dt " or "s =ut +(1)/(2) at^2" ""............."(2)`
Velocity -displacement relation
(iii) The accelaration is given by the first derivative of velocity with respect to time.
`a= (dv)/(dt)=(dv)/(ds)(ds)/(dt)= (dv)/(ds)v`
[ since `(ds)/(dt)=v`] where s is distance traversed]
This is rewritten as `a=(1)/(2) (dv^2)/(ds)" or " ds = (1)/(2a) d(v^2)`
Integrating the above equation, using the fact when the velocity changes from `u^2" to " v^2`, displacement changes from 0 to s, we get
`int_(0)^(s) ds = int_(u)^(v) (1)/(2a)d (v^2)`
`:. S= (1)/(2a) (v^2-u^2)`
`:. v^2=u^2+2as" ""............."(3)`
We can also derive the displacement 's' in terms of initial velocity 'u' and final velocity v. From equation 1, we can write
`at = v-u`
Substitute this in equation 2, we get
`S= ut +(1)/(2)(v-u)t`
`S= ((u+v)t)/(2)" "".............."(4)`
The equations 1,2,3 and 4 are called kinematic equations of motion, and have a wide variety of practical applications.
Kinematic equations
`v= u+at`
`S= ut +(1)/(2) at^2`
`v^2= u^2+2as`
`s=((u+v)t)/(2)`
It is to be noted that all these kinematic equations are valid only if the motion is in a straight line with cosntant acceleration. For circular motion and oscillatory motion these equations are not applicable.