How do we know that the escape velocity formula is what it is? How was it determined?
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So, we’re going to start with some constants. How we got them isn’t important for this topic.
* G: the universal *gravitational constant*
* M: mass of the Earth
* m: mass of the escaping body
At a given distance *r* from the center, the body feels a force pulling it towards the center:
F = G ( (M*m) / r^2 )
The work (force exerted over distance) needed to move the body over a distance *dr* is then:
dW = F * dr = G ( (M*m) / r^2 ) * dr
In order to extend this work from the surface of the planet r0 to arbitrarily far away from Earth, we have to integrate:
>dW = integral from r0 to infinity of:
G ( (M*m) / r^2 ) * dr
The dr term falls away, and we’re left with:
W = G ( (M*m) / r0 )
In order for the body to be able to do this amount of work, it must have exactly this amount of kinetic energy. Recall the formula for kinetic energy, and we have
1/2 mv^2 = G ( (M*m) / r0 )
Here we rearrange terms to isolate the velocity, v, and we arrive at the formula for escape velocity:
v = sqrt (2GM / r0) = sqrt (2gr0)
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