💬 Gravity:

$$ F_g=G\frac{m_1m_2}{r^2} $$

$G=6.67\;10^{-11}\;\text{m}^3\text{kg}^{-1}\text{s}^{-2}$

$F_g\propto \frac{1}{r^2}$

💪 The Gravitational force an acceleration g:

$$ F=mg \;\text{ ; }\; g=\frac{GM_{earth}}{R^2_{earth}} $$

💼 Work:

• for constant $F$:

$$ W=Fs=mgs=K $$

• otherwise:

$$ W=\int F\text{d}s=\int G\frac{m_1m_2}{r^2}\,\text{d}r $$

• if something is falling a distance $s$ from $r_1$ to $r_2$ with $r_1=r_2+s$ then:

$$ W=-\left[G\frac{m_1m_2}{r}\right]^{r_1}_{r_2}=Gm_1m_2\left(\frac{1}{r_2}-\frac{1}{r_2+s}\right) $$

• asuming a fall from infinity we get ($r_1=\infin$):

$$ W=G\frac{m_1m_2}{r^2}=\frac{1}{2}m_1v^2 \;\text{ ; }\; E_p=-G\frac{m_1m_2}{r} $$

⌛ Constants:

$$ v_{esc}=\sqrt{\frac{2GM}{r}} $$

$$ v_{orbit}=\sqrt{\frac{GM}{r}}=\frac{1}{\sqrt{2}}v_{esc} $$