$$a=(0,g)$$
$$v=\int adt =(0,gt)+(v_0^1,v_0^2)=(v_0^1,v_0^2+gt)$$
$$s=\int vdt =(v_0^1t,\ \ v_0^2t+\dfrac{1}{2}gt^2)$$
$$s=(Rcos\theta,Rsin\theta)= (Rcos\omega t,Rsin\omega t)$$
$$|s|=R$$
$$v=\dot{s}=\dfrac{ds}{dt}=(-\omega Rsin\omega t,\omega Rcos\omega t)$$
$$|v|=\omega R$$
$$a=\dot{v}=\dfrac{dv}{dt}=(-\omega^2 Rcos\omega t,-\omega^2 Rsin\omega t)$$
$$|a|=\omega^2 R$$
$$z=R(cos\theta,isin\theta)=Re^{i\theta}$$
$$v=\dot{z}=\dfrac{dz}{dt}=Re^{(i\theta)}i\omega =i\omega z$$
$$a=\dot{v}=\dfrac{dv}{dt}=i\dot{\omega }z+i\omega \dot{z} =i\beta z-\omega ^2z=w^2z$$
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