# Graviational force and escape velocity

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Do gravitational force and escape velocity have a direct and constant relationship, specifically is the gravitational force at the schwarzild radius the same for any body? If so what is it?

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Escape velocity is given by $v_e = \sqrt{\frac{2GM}{r}}$ (https://en.wikipedia.org/wiki/Escape_velocity)

Where M is the mass of the planet (or whatever) and r is its radius. If you substitute the Schwarzschild radius ($r_s = \frac{2 G M}{c^2}$) for r then you will find the result comes to c. (https://en.wikipedia.org/wiki/Schwarzschild_radius)

The equation for gravitational force is given by $F = G \frac{m M}{r^2}$. As you can see, this decreases with r2 rather than square root of r for escape velocity. It also depends on the mass (m) of the object. (https://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation)

So escape velocity and gravitational force are related but not in a direct way.

The acceleration due to gravity (g) is independent of the mass of the object (because acceleration is force/mass):  $g = G \frac{M}{r^2}$

If we substitute this into the equation for escape velocity, we end up with:  $g = \frac{{v_e}^2}{2 r}$

p.s. I wouldn't be too surprised if someone comes along and points out that I have made a mistake there somewhere...

Edited by Strange

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Thank you, you always make things pretty clear...

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