\magnification=\magstep1

$$
d \quad = \quad {{1}\over{2}} \thinspace a \thinspace t^2
$$

$$
d \quad = \quad (x_f - x_i) \quad = \quad v_i \thinspace t  
        \ + \ {{1}\over{2}} \thinspace a \thinspace t^2
$$


$$
{\vec{v}} \quad = \quad  { {d\vec{x}} \over {dt} }
$$

$$
{\vec{a}} \quad = \quad  { {d\vec{v}} \over {dt} } \quad 
        = \quad { {d^2\vec{x}} \over {dt^2} } \quad 
$$

$$
{\vec{a}} \quad = \quad  { {d\vec{v}} \over {dt} } \qquad 
     \Longrightarrow \qquad
     {\vec{v}} \quad = \quad \int{ {\vec{a}} \ dt }
$$

$$
 {\vec{v}} \quad = \quad {\vec{a}}  \int{ dt }
$$

$$
  {\vec{v}} \quad = \quad {\vec{v}_i} \ + \ {\vec{a}} \thinspace t
$$

$$
{\vec{v}} \quad = \quad  { {d\vec{x}} \over {dt} } \qquad 
     \Longrightarrow \qquad
     {\vec{x}} \quad = \quad \int{ {\vec{v}} \ dt }
$$


$$
  {\vec{x}} \quad = \quad \int{ \left( {\vec{v}_i} \ + \ {\vec{a}} \thinspace t  \right) dt }
$$

$$
  {\vec{x}} \quad = \quad \int{ {\vec{v}_i} \ dt } \quad + \quad \int{ {\vec{a}} \thinspace t \ dt }
$$

$$
  {\vec{x}} \quad = \quad {\vec{x}_i} \quad + \quad {{\vec{v}_i} \ t}  \quad + 
         \quad { {{1}\over{2}} \vec{a} \ t^2  }
$$

$$
  {\vec{x}} \quad = \quad {\vec{x}_i} \quad + \quad  
        { {{1}\over{2}} ( {\vec{v}_i} + {\vec{v}_f} ) \ t }
$$

$$
  {{v}_f^2}  -  {{v}_i^2} \quad = \quad 
          2 \ {\vec{a}} \ ( {\vec{x}_f} - {\vec{x}_i} )
$$
 

$$
v(t) \quad = \quad 10\ \left( {\rm {{mph}\over{hour}}} \right)  \ t  
  \quad = \quad
                10\ \left( {\rm {{mile}\over{hour^2}}} \right) \thinspace 
                       t 
$$


$$
(x - x_i) \quad = \quad v_i \thinspace t \ + \ {{1}\over{2}} \thinspace a t^2
$$

$$
    d     \quad = \quad  0 \ + \ {{1}\over{2}} \thinspace a t^2
$$

$$
t^2 \quad = \quad \sqrt{ {{2 d}\over{a}} } 
    \quad = \quad 
         \sqrt{  {{2 \thinspace (75\ {\rm mile})}\over{10\ {\rm mile/hour^2}}} }
    \quad = \quad 
          3.9 \ {\rm hour}
$$





\bye
