Earth's escape velocity

You'd be surprised how many intriguing Physics results are on the borderline between high school Maths and College Maths. This one is one of my favorites, mainly because it is actually applied (probably) in rocket science and it gives the feeling of greatness to anyone calculating/understanding it.

Before we start, let me give you some formulas:
The force of attraction between two bodies of masses M and m respectively, with a distance R between their center of masses, is $$F=\frac{GMm}{R^2}$$ (I actually managed to properly write fractions!)
G is the constant of gravitation, which is roughly equal to \(6.7*10^{-11} Nkg^{-2}m^2\). The mass of the Earth is \(5.972*10^{24}kg\), but, to make the job of the calculator easier, we shall use \(6*10^{24}kg\). The radius of the Earth is \(6,4*10^{6}m\). Now that we know these formulas, we can get to the task at hand.

The net force required to leave the Earth is, obviously, equal to the attraction force. Because an object is incredibly small compared to the radius of a planet, we shall consider the distance between the object and planet equal to the radius of the planet.

In order to leave the planet, one must exert work with distance reaching infinity, and, thus, the work needed to escape the planet is $$\int_R^⧞ \frac {MmG}{R^2} dr=\frac {MmG}{R}=\frac{mv^2}{2}$$.

Thus, we get that \(v^2=\frac {2MG}{R}\), so \(v={(2MG)^{1/2}}{R^{1/2}}\). By plugging in the mentioned constants, we get that the speed is about 11.20825 kilometers per second. That's roughly 40338 kilometers per hour.

You'd think that such a speed can be achieved only by incredibly complex pieces of technology with highly complex engines and complicated patterns, yet the first object to reach escape velocity is a steel plate covering a nuclear test back in 1957. It is unknown if it did actually leave Earth's atmosphere, as it would have probably boiled due to the high speed, but one thing is sure: no one found its debris. Who knows, maybe the first man-made creation the aliens find won't be one of the Voyager probes, but rather a random steel plate floating randomly through the Solar System.

Cris.