
\magnification=\magstep1


$$
\eqalign{
 \sum F_x \quad &= \quad m a_x \cr
 \sum F_y \quad &= \quad m a_y \cr
}
$$

$$
\eqalign{
{\rm KE}_i \ + \ {\rm PE}_i \quad &= \quad {\rm KE}_f \ + \ {\rm PE}_f  \ - \ {\rm Work}_{\rm noncons}  \cr
}
$$

$$
\eqalign{
{\rm KE}_i \ + \ {\rm PE}_i \quad &= \quad {\rm KE}_f \ + \ {\rm PE}_f  \ - \ 0 \cr
}
$$

$$
3600 \thinspace J \ + \ 39200 \thinspace J \quad = 
       \quad {{1}\over{2}} m v_f^2 \ + \ 23520 \thinspace J \ - \ 0 \thinspace  J
$$

$$
F_{\rm fric} \quad = \quad \mu_k m g \cos(\theta) \quad = \quad 76.1 \thinspace N
$$

$$
{\rm W_{\rm fric}} \quad = \quad \int { \vec{F}_{\rm fric} \cdot \vec{dx} }
$$

$$
3600 \thinspace J \ + \ 39200 \thinspace J \quad = 
       \quad {{1}\over{2}} m v_f^2 \ + \ 23520 \thinspace J \ - \ (-1141.4 \thinspace J)
$$

$$
{\rm force\ of\ gravity: } \qquad \vec{F}_g \quad = \quad - m g \thinspace \hat{j}
$$


$$
{\rm grav\ potential\ energy: } \qquad {\rm GPE}(B) - {\rm GPE}(A) 
          \quad = \quad 
                   - \int_A^B \vec{F}_g \cdot \vec{dx}
$$

$$
\vec{F} \quad = \quad - \thinspace {{ \partial({\rm PE})\over{\partial \vec{x}}}}
$$

$$
\vec{F} \quad \simeq \quad - \thinspace {{({\rm change\ in\ PE})}
                    \over{({\rm displacement})} }  \qquad {\rm in\ dir\ of\ displacement}
$$

$$
{\rm PE}(x) \quad = \quad 3.2 \times 10^{3} \thinspace
                       x^2 \thinspace\thinspace {{\rm J}\over{\rm m^2}}
$$

$$
{\rm KE}_i \ + \ {\rm GPE}_i \quad = \quad {\rm KE}_f \ + \ {\rm GPE}_f
$$

$$
{{1}\over{2}} m v_i^2 \ + \ m g h_i \quad = \quad 
                 {{1}\over{2}} m (0)^2 \ + \ m g h_f
$$

$$
{{1}\over{2}} m v_i^2 \ + \ {\rm GPE}_i \quad = \quad 
                      \ 0 \ \   + \ {\rm GPE}_f
$$


$$
{{1}\over{2}} m v_i^2 \quad = \quad {\rm GPE}_f \ - \ {\rm GPE}_i
$$



$$
{\rm GPE}_f - {\rm GPE}_i \quad = \quad - \thinspace \int_{h_i}^{h_f} 
         {{G M_E m}\over{y^2}} ({-\hat{j}}) \cdot \vec{dy}
   \quad = \quad 
         \int_{h_i}^{h_f} {{G M_E m}\over{y^2}} {dy}
$$


$$
{\rm GPE}_f - {\rm GPE}_i \quad = \quad 
         {{G M_E m}\over{h_i}} \ - \ {{G M_E m}\over{h_f}} 
$$


$$
{{G M_E m}\over{h_f}}  \quad = \quad
         {{G M_E m}\over{h_i}} \ - \ {{1}\over{2}} m v_i^2 
$$

$$
h_f \quad = \quad {{1}\over{  {{1}\over{h_i}} \ - \ {{v^2}\over{2 G M_E}} }}
$$


$$
\eqalign{
{\rm GPE}_f - {\rm GPE}_i \quad &= \quad 
         {{G M_E m}\over{h_i}} \ - \ {{G M_E m}\over{h_f}}  \cr
\phantom{.} \cr
                                &= \quad 
         {{G M_E m}\over{h_i}} \ - \ {{G M_E m}\over{\infty}}  
       \quad = \quad     
         {{G M_E m}\over{h_i}} \cr
}
$$


$$
{{1}\over{2}} m v_{esc}^2 \quad = \quad {{G M_E m}\over{h_i}} 
$$

$$
v_{esc} \quad = \quad \sqrt{  {{2 G M}\over{R}} }
$$

$$
R \quad = \quad  {{2 G M}\over{c^2}} 
$$



\end
