\magnification=\magstep1



$$
y \quad = \quad x^2
$$

$$ 
{ {dy}\over{dx} } \quad = \quad  2x
$$

$$
{ {dv}\over{dt} } \quad = \quad a \quad =  \quad -g 
$$

$$
v \quad = \quad at + v_0 \quad = \quad -gt + v_0
$$

$$
{ {dy}\over{dt} } \quad = \quad v \quad = -gt  \thinspace + \thinspace  v_0
$$

$$
y \quad = \quad - {{1}\over{2}} g t^2 \thinspace + \thinspace v_0 t 
        \thinspace + \thinspace y_0
$$

$$
{ {dv}\over{dt}} \quad = \quad -g \ + \ Kv^2
$$

$$
{ {d^2y}\over{dt^2} } \quad = \quad { {dv}\over{dt} } \quad = 
        \quad a \quad = \quad -g \quad = \quad -9.8 {\rm \ m/s^2}
$$

$$
v(t) \quad = \quad v_0 \ - \ gt
$$

$$
v(t) \quad = \quad 0 \ - \ gt
$$

$$
y(t) \quad = \quad y_0 \ + \ v_0 t \ - \ {{1}\over{2}} g t^2
$$

$$
y(t) \quad = \quad H \  - \ {{1}\over{2}} g t^2
$$

$$
{{dv}\over{dt}} \quad = \quad -g
$$

$$
v(t) \quad = \quad v(t_0) \ + \ v' (t - t_0) \ + \ {{1}\over{2}} v'' {(t - t_0)}^2
      \ + \ ...
$$

$$
v(t) \quad = \quad v(t_0) \ + \ v' (t - t_0) 
$$


$$
v_{i+1} \quad = \quad v_i \ + \ v'_{i} (t_{i+1} - t_i)
$$

$$
dt \quad = \quad t_{i + 1} \ - \  t
$$

$$
v' \quad = \quad {{dv}\over{dt}} \quad = \quad -g
$$

$$
v'' \quad = \quad {{d^2 v}\over{dt^2}} \quad = \quad 0
$$

$$
v(t) \quad = \quad v(t_0) \ + \ (-g)(t - t_0) \ + \ 0 \ + \ 0 \ ...
$$

$$
v(t) \quad = \quad v(t_0) \ + \ (-g)(t - t_0) 
$$

$$
v_{i+1} \quad = \quad v_i \ + \ (-g)(t_{i+1} - t_i) 
$$

$$
y(t) \quad = \quad y(t_0) \ + \ y'(t - t_0) \ + \ {{1}\over{2}} y''{(t - t_0)}^2 \ + \ ...
$$

$$
\eqalign{
y' \quad &= \quad v \cr
y'' \quad &= \quad a \quad = \quad -g \cr
y''' \quad &= \quad 0 \cr
}
$$


$$
y_{i+1} \quad = \quad y_i \ + \ v (t_{i+1} - t_i)
$$

$$
{\rm error \ } \quad = \quad {{1}\over{2}} \thinspace g {(t_{i+1} - t_i)}^2
$$

$$
y_{i+1} \quad = \quad y_i \ + \ v(t_{i+1} - t_i) \ - \ 
            {{1}\over{2}} g {(t_{i+1} - t_i)}^2 
$$



\bye



