\magnification=\magstep2

$$
\vec{F} = m \vec{a}
$$

$$
{ {\int_{t_1}^{t_2} {\vec{F} \thinspace dt} } = { {\int_{t_1}^{t_2} { m \vec{a} \thinspace dt } }}   }
$$

$$
{ {\int_{t_1}^{t_2} {\vec{F} \thinspace dt} } = { m {\int_{t_1}^{t_2} { \vec{a} \thinspace dt } }}   }
$$

$$
{ {\int_{t_1}^{t_2} {\vec{F} \thinspace dt} } = { m ( { \vec{v}_{2} - \vec{v}_1 } ) }}   
$$

$$
{ {\int_{t_1}^{t_2} {\vec{F} \thinspace dt} } = { ( m \vec{v}_{2} ) - ( m \vec{v}_1 ) }}   
$$

$$
{ {\int_{t_1}^{t_2} {\vec{F} \thinspace dt} } = { \vec{p}_{2}  -  \vec{p}_1  }}   
$$




$$
{ { d \vec{p} } \over { dt } }  =  \sum \vec{F}_{ext}
$$

$$
{ { d \vec{p}_{tot} } \over { dt } }  =  0 \quad 
                       \Rightarrow \quad \vec{p}_{tot} = {\rm constant}
$$

$$
 \vec{p}_{B} - \vec{p}_{A}   =  \int_A^B {{F}_{ext} \thinspace dt }
$$


$$
 {\rm final\ \ }  {p}_x = {\rm initial\ \ } {p}_x
$$
$$
 {\rm final\ \ }  {p}_y = {\rm initial\ \ } {p}_y 
$$
$$
 {\rm final\ \ }  {p}_z = {\rm initial\ \ } {p}_z 
$$

$$
  \vec{p}_{final} = \vec{p}_{initial}
$$

$$
{\int_{t_1}^{t_2} {\vec{F} \thinspace dt} } \quad =  \quad
           {\vec{F}}_{\rm avg}  { {\int_{t_1}^{t_2} \thinspace {dt} } } 
                     \quad = \quad
           {\vec{F}}_{\rm avg} \thinspace  (t_2 - t_1)
$$



\bye
