
\magnification=\magstep2


$$
y'(t) \quad = \quad f(t) \qquad {\rm (but\ not\ a\ function\ of\ y)}
$$

$$
y'(t) \quad = \quad - (9.8 {{m}\over{s^2}}) \thinspace t
$$

$$
y_{i + 1} \quad = \quad y_i \ + \ y'_i \thinspace dt
$$

$$
y_{i + 1} \quad = \quad y_i \ + \ {{1}\over{2}} (y'_i + y'_{i+1}) \thinspace dt
$$

$$
y_{i + 1} \quad = \quad y_i \ + \ {{(y'_i \ + \  4 y'_{i+1/2} \ + \ y'_{i+1})}\over{6}} \thinspace dt
$$


$$
y'(t) \quad = \quad f(y, t, a, b, ...) 
$$

$$
\eqalign{
{ {dv_y}\over{dt} } \quad &= \quad - {{G M} \over{y^2}} \thinspace
              { \left ({ y \over {|y|} } \right ) } \cr
{ {dy}\over{dt} } \quad &= \quad v_y \cr
}
$$

$$
\eqalign{
 t \quad &= \quad 0 \cr
 y \quad &= \quad 10 {\rm \ m} \cr
 v_y \quad &= \quad 0 {\rm \ m/s} \cr
}
$$

$$
\eqalign{
{\rm predict\ \ } \tilde{y}'_{i+1} \quad & = \quad y'_i \ + \ 
                           y''_i \thinspace dt \cr
      &= \ \ \ \ 0 \ - \ 0.667 {\rm \ m/s^2} (1{\rm \ s})  \cr
      &= \ -0.667 {\rm \ m/s^2} \cr
}
$$


$$
\eqalign{
{\rm predict\ \ } \tilde{y}_{i+1} \quad & = \quad y_i \ + \ 
                           y'_i \thinspace dt \cr
      &= \ 10{\rm \ m} \ + \ 0 {\rm \ m/s} (1{\rm \ s})  \cr
      &= \ 10 {\rm \ m} \cr
}
$$

$$
\eqalign{
{\rm predict\ \ } \tilde{y}''_{i+1} \quad &= \quad 
                           - {{G M}\over{ {(\tilde{y}_{i+1})}^2 } } \cr
         &= \quad - 0.667 {\rm \ m/s^2} \cr
}
$$


$$
\eqalign{
{y}'_{i+1} \quad & = \quad y'_i \ + \ 
                 {{1}\over{2}} (y''_i + \tilde{y}''_{i+1}) \thinspace dt \cr
      &= \ \ \ \ 0 \ - \ 0.667 {\rm \ m/s^2} (1{\rm \ s})  \cr
      &= \ -0.667 {\rm \ m/s^2} \cr
}
$$

$$
\eqalign{
{y}_{i+1} \quad & = \quad y_i \ + \ 
                 {{1}\over{2}} (y'_i + \tilde{y}'_{i+1}) \thinspace dt \cr
      &= \ 10 {\rm \ m} \ + \ {{1}\over{2}} ( 0 {\rm \ m/s} \ - \ 
                          0.667 {\rm \ m/s}) (1 {\rm \ s}) \cr
      &= \ 9.66 {\rm \ m} \cr
}
$$


\bye
