g
9.81
0
r
R
∞
P
r
x
dx
𝐺𝑀𝑚
𝑈=−
𝑟
𝐹=
𝑚𝑣 2
𝑟
𝑎𝑛𝑑 𝐹 =
Equalise these two equations
𝑚𝑣 2 𝐺𝑀𝑚
= 2
𝑟
𝑟
Rearranging gives:
𝑣2 =
𝐺𝑀
𝑟
Total energy of a satellite is GPE + KE
𝐺𝑃𝐸 + 𝐾𝐸 = −
𝐺𝑀𝑚
𝑟
1
+ 2 𝑚𝑣 2 replace 𝑣 2 from 𝑣 2 =
𝐺𝑀
𝑟
By Conservation of Energy,
Initial 𝐾𝐸 + Initial 𝐺𝑃𝐸 = Final 𝐾𝐸 + Final 𝐺𝑃𝐸 :
1
2
𝑚𝑣 2 + (−
𝐺𝑀𝑚
𝑟
)
=
0
+
0
So, let’s say the escape velocity is 𝑣𝐸 then, 𝑣 2 =
Thus, the escape speed 𝑣𝐸 = √
2𝐺𝑀
𝑟
2𝐺𝑀
𝑟
𝑇2 𝛼 𝑟3
𝐺𝑀𝑚
𝑟2