\begin{aligned}∑ F_{x} & = ma_{x} \\ ∑ F_{y} & = ma_{y} \end{aligned}

KE_{i} + PE_{i} = KE_{f} + PE_{f} − Work_{noncons}

KE_{i} + PE_{i} = KE_{f} + PE_{f} − 0

3600 J + 39200 J = \frac{1}{2}mv_{f}^{2} + 23520 J − 0 J

F_{fric} = μ_{k}mg cos(θ) = 76.1 N

W_{fric} = ∫ F⃗_{fric} · d⃗x

3600 J + 39200 J = \frac{1}{2}mv_{f}^{2} + 23520 J − (−1141.4 J)

force of gravity :  F⃗_{g} = −mg ĵ

grav potential energy : GPE(B) − GPE(A) = − ∫_{A}^{B} F⃗_{g} · d⃗x

F⃗ = − \frac{∂(PE)}{∂x⃗}

F⃗ ≃ − \frac{(change in PE)}{(displacement)} in dir of displacement

PE(x) = 3.2 × 10^{3} x^{2} \frac{J}{m^{2}}

KE_{i} + GPE_{i} = KE_{f} + GPE_{f}

\frac{1}{2}mv_{i}^{2} + mgh_{i} = \frac{1}{2}m(0)^{2} + mgh_{f}

\frac{1}{2}mv_{i}^{2} + GPE_{i} = 0 + GPE_{f}

\frac{1}{2}mv_{i}^{2} = GPE_{f} − GPE_{i}

GPE_{f} − GPE_{i} = − ∫_{h_{i}}^{h_{f}} \frac{GM_{E}m}{y^{2}}(−ĵ) · d⃗y = ∫_{h_{i}}^{h_{f}} \frac{GM_{E}m}{y^{2}}dy

GPE_{f} − GPE_{i} = \frac{GM_{E}m}{h_{i}} − \frac{GM_{E}m}{h_{f}}

\frac{GM_{E}m}{h_{f}} = \frac{GM_{E}m}{h_{i}} − \frac{1}{2}mv_{i}^{2}

h_{f} = \frac{1}{\frac{1}{h_{i}} − \frac{v^{2}}{2GM_{E}}}

\begin{aligned}GPE_{f} − GPE_{i} & = \frac{GM_{E}m}{h_{i}} − \frac{GM_{E}m}{h_{f}} \\ & = \frac{GM_{E}m}{h_{i}} − \frac{GM_{E}m}{∞} = \frac{GM_{E}m}{h_{i}} \end{aligned}

\frac{1}{2}mv_{esc}^{2} = \frac{GM_{E}m}{h_{i}}

v_{esc} = \sqrt{\frac{2GM}{R}}

R = \frac{2GM}{c^{2}}