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Hey, I'm a first year Astrophysics student, and in revising the Schwarzchild radius, I wanted to derive it and so I started by deriving the escape velocity from first principles, then rearrange to get the Schwarzchild Radius. Child's play, really.

However, depending from where you come from, either from energy or rotational motion, you end up being a factor of [tex]\sqrt{2}[/tex] out:

I am ignoring vector notation for speed

[tex]F = \frac{GMm}{r^{2}} = ma[/tex]

[tex]a = \frac{GM}{r^{2}}[/tex]

However,

[tex]a = \frac{v^{2}}{r}[/tex]

Therefore,

[tex]\frac{v^{2}}{r} = \frac{GM}{r^{2}}[/tex]

Rearranging for v gives

[tex]v = \sqrt{\frac{GM}{r}}[/tex]

We can assume that by units,

[tex]E = \frac{1}{2}mv^{2} = \frac{GMm}{r}[/tex]

[tex]E = \frac{1}{2}v^{2} = \frac{GM}{r}[/tex]

[tex]E = v^{2} = \frac{2GM}{r}[/tex]

[tex]v = \sqrt{\frac{2GM}{r}}[/tex]

Are my mathematics skills not up to scratch, or is it that for two equally valid derivations, we get two equally valid equations, that mean the same thing, but are not equal to one another?

However, depending from where you come from, either from energy or rotational motion, you end up being a factor of [tex]\sqrt{2}[/tex] out:

**First Derivation, from Newtonian Mechanics**I am ignoring vector notation for speed

[tex]F = \frac{GMm}{r^{2}} = ma[/tex]

[tex]a = \frac{GM}{r^{2}}[/tex]

However,

[tex]a = \frac{v^{2}}{r}[/tex]

Therefore,

[tex]\frac{v^{2}}{r} = \frac{GM}{r^{2}}[/tex]

Rearranging for v gives

[tex]v = \sqrt{\frac{GM}{r}}[/tex]

**Second Derivation, from Energy**We can assume that by units,

[tex]E = \frac{1}{2}mv^{2} = \frac{GMm}{r}[/tex]

[tex]E = \frac{1}{2}v^{2} = \frac{GM}{r}[/tex]

[tex]E = v^{2} = \frac{2GM}{r}[/tex]

[tex]v = \sqrt{\frac{2GM}{r}}[/tex]

Are my mathematics skills not up to scratch, or is it that for two equally valid derivations, we get two equally valid equations, that mean the same thing, but are not equal to one another?

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