An object rotates with angular acceleration, $\alpha$, defined as
$$
\alpha=\frac{\mathrm{d} \omega}{\mathrm{d} t}
$$
where $\omega$ is the angular velocity. Given that when $t=0, \omega=\omega_{0}$, show that
$$
\omega=\omega_{0}+\alpha t
$$
Also the angular velocity $\omega$ is defined as
$$
\omega=\frac{\mathrm{d} \theta}{\mathrm{d} t}
$$
where $\theta$ is the angular displacement. Given that when $t=0, \theta=0$ show that
$$
\theta=\omega_{0} t+\frac{1}{2} \alpha t^{2}
$$