When an object reaches escape velocity it will eventually travel an infinite distance away from the planet, star, or moon that is exerting a gravitational pull on it and never fall back. That is of course if nothing else happens, such as an astronaut turning his spacecraft around and coming back. This velocity needed to escape to an infinite distance is an instantaneous velocity; this means it does not need to be maintained; the object will gradually slow down but will never stop.
What Determines Escape Velocity?
The escape velocity needed depends, of course, on what we are escaping from; in other words, the mass of that body. It also depends on how far we are away from that body to begin with. Armed with just those two facts, we can calculate this figure for any body in the universe.
Calculating Escape Velocity
Let’s imagine, for example, that we drop an object from an infinite height above the Earth to a distance r from the centre of Earth. Now at that position in space, imagine there is a gigantic trampoline. The object would hit that trampoline and rebound with the same velocity or escape velocity needed to get back to the point an infinite distance away (if the trampoline was perfect). Of course, this experiment would take an infinite amount of time, and would be quite costly.
Instead, let us conduct a mathematical experiment. In the example we have just looked at, the gravitational or potential energy of that object we dropped turned into kinetic energy, and from that equation we can calculate the final or escape velocity. But it’s not quite that simple; is the acceleration constant? No, the gravitational force increases as the object gets nearer to the Earth, it is never constant. We need to use calculus therefore to arrive at a formula to calculate the escape velocity.
Integrating Gravity
To calculate the total work done or the gravitational energy used to speed up the falling object we must add up all the force times distance segments acting on our object (look at figure 1). The final kinetic energy of the object is equal to the area in the diagram that is below the curve, and this can be calculated by integrating this function between infinity and r.
Gravitational Force, F = GMm/R² (G is the gravitational constant; M is the mass of the Sun, planet or moon; m is the mass of the object; R is the distance between the centre of both masses).
The sum of all force x distance segments between infinity and r :
Kinetic Energy = sum of all (force x small change in distance).
The integration interval is between infinity and r.
mv²/2 = GMm ∫ (1/R²) dR = GMm(1/r) – GMm(1/∞)
As 1/∞ = 0 we can write:
mv²/2 = GMm(1/r) – 0 = GMm/r
v²/2 = GM/r
v² = 2GM/r
Escape Velocity, v = √(2GM/r)
Escaping Earth
Let’s find what this velocity is for, say, a satellite at a height of 100 miles or 160 kilometres. That is the radius of the Earth + (160 x 10³) metres = (6371 x 10³) + (160 x 10³) = 6531 x 10³ metres. (Remember mass and distance figures must be in kilograms and metres because G is a metric constant).
Escape Velocity, v = √(2 x 6.672 x 10‾ ¹¹ x 597.8 x 10²² x (1/6531) x 10‾³).
v = √(.0122 x 10¹º)
v = 11 x 10³ metres per second or 11 kilometres per second.
That is 6.875 miles per second or 24,750 miles per hour. Most sources give Earth's escape velocity at 11.2 kilometres per second, but this is calculated from ground level and would result in immediate destruction in Earth's atmosphere for any spacecraft. Another interesting fact comes from this equation, and it is that an object’s escape velocity also equals its circular orbital velocity multiplied by √2.
Stronger Forces
The only problem with the concept of escape velocity is that it assumes the object and the body it is escaping from are all that exist in the universe. If a spacecraft reached the Earth's escape velocity, it would still be under the influence of a much larger body. In the solar system, the Earth is dwarfed by the Sun. So to escape our solar system, a spacecraft would need to reach the escape velocity of the Sun at our position in space. We can find that velocity by multiplying Earth's circular orbital velocity by the square root of 2. We arrive at a final velocity of 94,747 miles per hour to escape our solar system.
References:
- Geological Science by Andrew McLeish published by Nelson Thornes 2001.
- Fundamental Astronomy by Hannu Karttunen, Pekka Kroger, Heikki Oja, Markku Poutanen, Karl J. Donner. Published by Springer, 2007.
Bruce Bowden - Bruce Bowden writes about science and other topics. His book Strange Harbinger is a science fiction mixture of comedy and drama.
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