How to calculate the escape velocity?
The advent of SpaceX seems to have given a new life to public interest for space explorations. In light of this, my first log is about calculating the escape velocity.
It is known from school course of physics that Ep = mgh and Ek = 2-1mV2, where m - mass of the object, g - acceleration due to gravity, h - altitude and V - velocity of the object. When we want to calculate maximum altitude, velocity at impact or its change we use this:
mgh = 2-1mV2
This is however insufficient to calculate the escape velocity because when the escape conditions are met (h = ∞), we get V = (2g∞)1/2 = (∞)1/2 which is impossible to calculate. The reason behind this is g which we treat as constant because the altitude is so small with respect to radius of Earth making any change in g too insignificant to consider. It becomes slightly more challenging to find the product of g and h when g is the function of h. Therefore we need integration. First step is to specify g(h).
According to the Universal Law of Gravity:
mg = GmMr-2
Where G is Universal Gravitational Constant, M - mass of the celestial body and r - radius of the celestial body. Altitude (h) is equal to 0 at the surface level and is collinear to radius so it is the variable to be added to the radius:
g(h) = GM(r+h)-2
In order to better understand our goal it is good to draw a graph. Its purpose is purely illustrative to visualize the area under it, the one to be calculated.
So gh is equal to:
∫
h=∞
GM(r+h)-2dh
h=0
G and M are constants so they can be put outside. r is a constant as well but it is added to the variable and then the sum is put to -2 power which means it will stay inside.
GM
∫
h=∞
h=∞
|
h=0
|
(r+h)-2dh
=
-GM
(r+h)-1
=
GM
(r+h)-1
=
h=0
h=0
h=∞
= GM[(r+0)-1 + (r+∞)-1] = GM[r-1 + 0] = GMr-1
2-1V2esc = GMr-1; V2esc = 2GMr-1; Vesc = (2GMr-1)1/2
This formula is to be found after the search by “escape velocity” keywords although its appearance will be different. However it only represents a projectile fired from the surface. Space rockets reach the needed velocity at a significant altitude, so from our previous calculations:
Vesc = [2GM(r+h)-1]1/2
It is known from school course of physics that Ep = mgh and Ek = 2-1mV2, where m - mass of the object, g - acceleration due to gravity, h - altitude and V - velocity of the object. When we want to calculate maximum altitude, velocity at impact or its change we use this:
mgh = 2-1mV2
This is however insufficient to calculate the escape velocity because when the escape conditions are met (h = ∞), we get V = (2g∞)1/2 = (∞)1/2 which is impossible to calculate. The reason behind this is g which we treat as constant because the altitude is so small with respect to radius of Earth making any change in g too insignificant to consider. It becomes slightly more challenging to find the product of g and h when g is the function of h. Therefore we need integration. First step is to specify g(h).
According to the Universal Law of Gravity:
mg = GmMr-2
Where G is Universal Gravitational Constant, M - mass of the celestial body and r - radius of the celestial body. Altitude (h) is equal to 0 at the surface level and is collinear to radius so it is the variable to be added to the radius:
g(h) = GM(r+h)-2
In order to better understand our goal it is good to draw a graph. Its purpose is purely illustrative to visualize the area under it, the one to be calculated.
So gh is equal to:
∫
h=∞
GM(r+h)-2dh
h=0
G and M are constants so they can be put outside. r is a constant as well but it is added to the variable and then the sum is put to -2 power which means it will stay inside.
GM
∫
h=∞
h=∞
|
h=0
|
(r+h)-2dh
=
-GM
(r+h)-1
=
GM
(r+h)-1
=
h=0
h=0
h=∞
= GM[(r+0)-1 + (r+∞)-1] = GM[r-1 + 0] = GMr-1
2-1V2esc = GMr-1; V2esc = 2GMr-1; Vesc = (2GMr-1)1/2
This formula is to be found after the search by “escape velocity” keywords although its appearance will be different. However it only represents a projectile fired from the surface. Space rockets reach the needed velocity at a significant altitude, so from our previous calculations:
Vesc = [2GM(r+h)-1]1/2