Escape Velocity of Projectile fired Upwards

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Theorem

Let $P$ be a planet.

Let $P$ have an Acceleration Due to Gravity at its surface of $g$.

Let $P$ have a radius of $R$.


Then the escape velocity of $P$ is given by:

$V = \sqrt {2 g R}$


Proof

Let a projectile $B$ of mass $m$ be fired vertically upwards from the surface of $P$ at such a speed that it escapes the gravitational field of $P$ completely.

$F$ be the force exerted on the projectile by the gravitational field of $P$.

Let $x$ be the distance of $B$ from the surface of $P$ at time $t$.

We have:

$F = -\dfrac k {x^2}$

where $k$ can be calculated at the surface of $P$ as:

\(\displaystyle F\) \(=\) \(\displaystyle m g\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac k{R^2}\)
\(\displaystyle \implies \ \ \) \(\displaystyle k\) \(=\) \(\displaystyle R^2 m g\)


Thus:

$F = -\dfrac {m g R^2} {x^2}$

Let the $B$ be travelling away from $P$ with speed $v$ at time $t$.

Thus:

\(\displaystyle F\) \(=\) \(\displaystyle -m \frac {\mathrm d v} {\mathrm d t}\)
\(\displaystyle \implies \ \ \) \(\displaystyle -\frac {m g R^2} {x^2}\) \(=\) \(\displaystyle m v \frac {\mathrm d v} {\mathrm d x}\)

So:

$\displaystyle -\int \frac {R^2 g} {x^2} \, \mathrm d x = \int v \, \mathrm d v$

This gives:

$\dfrac {R^2 g} x = \dfrac {v^2} 2 + C$

Now when $x = R$, $v = v_0$, the launch velocity, which leads to:

$C = R g - \dfrac {v_0^2} 2$

and so:

$\dfrac {v^2} 2 = \dfrac {R^2 g} x - R g + \dfrac {v_0^2} 2$

For $\dfrac {v^2} 2 > 0$ we need:

$\dfrac {v_0^2} 2 > R g \left({1 - \dfrac R x}\right)$

whatever $x$ may be.

So when $x \to \infty$ we have:

$\dfrac {v_0^2} 2 > R g$

Hence:

$v_0 > \sqrt {2 R g}$

$\blacksquare$


Sources