y′(t) = f(t) (but not a function of y)

y′(t) = −(9.8\frac{m}{s^{2}} ) t

y_{i+1} = y_{i} + y′_{i} dt

y_{i+1} = y_{i} + \frac{1}{2}(y′_{i} + y′_{i+1}) dt

y_{i+1} = y_{i} + \frac{(y′_{i} + 4y′_{i+1/2} + y′_{i+1})}{6} dt

y′(t) = f(y, t, a, b, ...)

\begin{aligned}\frac{dv_{y}}{dt} & = −\frac{GM}{y^{2}} (\frac{y}{|y|}) \\ \frac{dy}{dt} & = v_{y} \end{aligned}

\begin{aligned}t & = 0 \\ y & = 10 m \\ v_{y} & = 0 m/s \end{aligned}

predict ỹ′_{i+1} = y′_{i} + y′′_{i} dt

= 0 − 0.667 m/s^{2}(1 s) = −0.667 m/s^{2}

predict ỹ_{i+1} = y_{i} + y′_{i} dt

= 10 m + 0 m/s(1 s) = 10 m

predict ỹ′′_{i+1} = −\frac{GM}{(ỹ_{i+1})^{2}}

= −0.667 m/s^{2}

y′_{i+1} = y′_{i} + \frac{1}{2}(y′′_{i} + ỹ′′_{i+1}) dt

= 0 − 0.667 m/s^{2}(1 s)

= −0.667 m/s^{2}

y_{i+1} = y_{i} + \frac{1}{2}(y′_{i} + ỹ′_{i+1}) dt

= 10 m + \frac{1}{2}(0 m/s − 0.667 m/s)(1 s)

= 9.66 m