d S(t)  dt 
= 
d (vt)  dt 
=  v 
d f(x))  dx 
= 
d ax  dx 
=  a 
Δ S(t)  Δ t 
Δ S(t)  Δ t 
= 
6  2 
=  3 
d S(t)  dt 
= 
d (vt)  dt 
=  v (equation 2) 
S'(t)  = 
d S(t)  dt 
=  speed 
S'(t)  = 
d S(t)  dt 
= at 
┌ 1 ┐ │ 3 │ └ 2 ┘ 
· 
┌ 0 ┐ │ 1 │ └ 0 ┘ 
= 3 
Δr(t)  Δt 
=  Δv(t) (equation 7) 
dr(t)  dt 
=  v(t) (equation 8) 
Δv(t)  Δt 
=  Δ a(t) (equation 9) 
dv(t)  dt 
= 
d^{2}r(t)  dt^{2} 
=  a(t) (equation 10) 
r(t)  =  r(t) r_{U}(t)  =  r(t) r_{U}(t) 
v(t)  = 
d r(t)  dt 
= 
d r(t)  r_{U}(t) dt 
+ 
d r_{U}(t)  r(t) dt 
(equation 8'  note the accent) 
d r_{U}(t)  dt 
r(t) = 
┌ 2cos(t) ┐ └ sin(t) ┘ 
v(t)  = 
dr(t)  dt 
= 
┌ 2sin(t) ┐ └ cos(t) ┘ 
dr(t)  dt 
= 
dx(t)  e_{x} dt 
+ 
dy(t)  e_{y} dt 
+ 
dz(t)  e_{z} dt 
S'(t)  = 
d S(t)  dt 
= 
d (vt)  dt 
=  v 
S'(t)  = 
d S(t)  dt 
= 
d  dt 
½ at^{2}  =  at  =  v 
dr(t)  dt 
=  v(t) 
dv(t)  dt 
= 
d^{2}r(t)  dt^{2} 
=  a(t) 
F_{1} = 
┌ 6 ┐ └ 0 ┘ 
F_{2} = 
┌ 12 ┐ └ 0 ┘ 
F_{R} = 
┌ 6 ┐ └ 0 ┘ 
+ 
┌ 12 ┐ └ 0 ┘ 
= 
┌ 6 + 12 ┐ └ 0 + 0 ┘ 
= 
┌ 6 ┐ └ 0 ┘ 
F_{1} = 
┌ 2 ┐ └ 5 ┘ 
F_{2} = 
┌ 1 ┐ └ 3 ┘ 
F_{R} = 
┌ 2 ┐ └ 5 ┘ 
+ 
┌ 1 ┐ └ 3 ┘ 
= 
┌ 2 + 1 ┐ └ 5 + 3 ┘ 
= 
┌ 1 ┐ └ 2 ┘ 
sin(α) = 
F_{p}  F_{g} 
a  = 
(v_{ AFTER}  v_{ BEFORE})  Δ t 
m a  =  m 
(v_{ AFTER}  v_{ BEFORE})  Δ t 
F  =  m 
(v_{ AFTER}  v_{ BEFORE})  Δ t 
F Δ t  =  m (v_{ AFTER}  v_{ BEFORE})  =  mv_{ AFTER}  mv_{ BEFORE} (equation 11) 
cos(α) = 
F_{P}  F 
v_{T}  = 
S  t 
= 
2πR  T 
(equation 22) 
ω  = 
2π  T 
= 
Δ θ  Δ t 
(equation 23) 
v_{T}  = 
2πR  T 
=  ωR  (equation 24) 
a_{T}  = 
v_{T}^{2}  R 
(equation 28) 
F_{T}  = 
mv_{T}^{2}  R 
(equation 29) 
F =  G 
M_{1} * M_{2}  r^{2} 
F =  6.673 * 10^{11} 
70 * 6*10^{24}  (6.38 x 10^{6})^{2} 
E_{pot, M1} =   G 
M_{1} * M_{2}  r 
Work of F from infinity to r = 
∫_{inf}^{r} G 
M_{1} * M_{2}

dr 
½ m * v_{esc}^{2}  +   G 
m * M  r 
= 0 
½ m * v_{esc}^{2}  =  G 
m * M  r 
½ v_{esc}^{2}  =  G 
M  r 
v_{esc}  =  √ ( 
2GM  r 
) 
F =  G 
M_{1} * M_{2}  r^{2} 
F =  6.673 * 10^{11} 
70 * 6*10^{24}  (6.38 x 10^{6})^{2} 
Δ Epot  =  E_{pot, orbit}  E_{pot, Earth}  = 
G*M_{E}*m  r_{orbit} 
 
G*M_{E}*m  r_{Earth} 