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\begin{document}

\frame{\frametitle{\b 1D Kinematics}
{\bf Oscillator (01):}
$$ \frac{d~}{dt}\,\sin(\omega\,t)= \omega\,\cos(\omega\,t)\,,~~~
   \frac{d~}{dt}\,\cos(\omega\,t)=-\omega\,\sin(\omega\,t)\,. $$
Uniformly {\bf increasing velocity (02)}. Acceleration:
$$ a = \frac{v_2-v_1}{t_2-t_1}\,.$$ 
New $t_0$, $t$, initial $v_0$:
$$ v_0=v_1+a\,(t_0-t_1)\,,~~x=v_0\,(t-t_0)+\frac{1}{2}\,a\,(t-t_0)^2\,.$$
{\bf Non-constant acceleration (04):}
$$ v(t) = \int dt\,a(t) + v_0\,,~~~x(t) = \int dt\,v(t) + x_0\,.$$
{\bf Free fall acceleration (06) A.}
Free fall with $v_0$ given:
$$ x(t) = x_0+v_0\,t-g\,\frac{t^2}{2}\,,~~x=x_0=0~~\Rightarrow~~
   0 = v_0 - g\,\frac{t}{2}~~\Rightarrow~~t\,.$$
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\frame{\frametitle{\b 1D Kinematics}
{\bf Free fall acceleration B.}
Greatest height:
$$ 0 = \frac{dx}{dt} = v_0 - g\,t_{\max}~~\Rightarrow~~
   t_{\max}=\frac{v_0}{g}\,,~~x_{\max} = x(t_{\max})\,.$$
{\bf Free fall acceleration C.}
Initial velocity $v_0$, time at first half velocity, $v=v_0/2$:
$$ v(t) = v_0 -g\,t\,,~~\frac{v_0}{2}=v_0-g\,t~~
   \Rightarrow~~t\,.$$
{\bf Free fall (07)} from heights $h$ and $h/2$:
$$ h = v_0\,t+\frac{1}{2}\,g\,t^2\,,~~\frac{h}{2} = \frac{1}{2}\,g\,t^2
   ~~\Rightarrow~~\frac{h}{2}=v_0\,t\,,~~t=\sqrt{\frac{h}{g}}~~~
   \Rightarrow~~v_0 =$$
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