Obtain equation `omega=omega_(0)+alphat` from first principle.

Explanation

The angular acceleration is uniform
`therefore (domega)/(dt)=alphaimpliesdomega=alphadt`
Integrating on both side
`intdomega=intalphadt`
`therefore omega=alphat+c`
where c is constant of integral
Now at `t=0, omega=omega_(0)`
`therefore omega_(0)=c`
`therefore` From above equation
`omega=alphat+omega_(0)`
`therefore omega=omega_(0)+alphat....(1)`
From this equation `theta=omega_(0)t+(1)/(2)alphat^(2)` can be obtain and `(d theta)/(dt)=omega_(0)+alphat`
`therefore d theta=omega_(0)dt+alphatdt`
Integrating on both side
`intd theta=underset(0)overset(t)intomega_(0)dt+underset(0)overset(t)intalphatdt`
`therefore theta=omega_(0)t=(alphat^(2))/(2)`
`therefore theta=omega_(0)t+(1)/(2)alphat^(2)...(2)`
From this eqn. `omega^(2)=omega_(0)^(2)+2alpha(theta-theta_(0))` can be obtain
From eqn. (1)
`t=(omega-omega_(0))/(alpha)`
Putting value of t in eqn. (2)
`theta-theta_(0)=omega_(0)((omega-omega_(0))/(t))+(1)/(2)alpha((omega-omega_(0))/(2))^(2)`
`theta-theta_(0)=(omega_(0)omega-omega_(0)^(2))/(alpha)+(omega^(2)-2omegaomega_(0)+omega_(0)^(2))/(2alpha)`
`=(2omegaomega_(0)-2omega_(0)^(2)+omega^(2)-2omegaomega_(0)+omega_(0)^(2))/(2alpha)`
`=(omega^(2)-omega_(0)^(2))/(2alpha)`
`therefore 2alpha(theta-theta_(0))=omega^(2)-omega_(0)^(2)`
`therefore omega^(2)=omega_(0)^(2)+2alpha(theta-theta_(0))....(3)`