Originally posted by DarkSaint85
I reckon it takes way more energy to smash a planet than to move it.
That depends on what you mean with moving and smashing.
If you by moving mean adding enough energy to break the Earth free from the gravitational pull of the Sun at its given distance. Then all you have to do is calculate the escape velocity of the Earth with respect to the Sun.
In which case we need to take a look at the formula for Newton's law of gravity;
and work:
If we combine the formulas we'll get;
or in integral form;
which we easily solve to be:
Now let's look at the relation between energy and (Newtonian) speed—which won't be a problem unless we're dealing with relativistic speeds, which we won't;
and we put it together with the previous formula:
The actually needs some explanation. m₁ and m₂ are the masses of the bodies that attract each other—in this case the Sun and the Earth—and we are interested in the movement of the Earth so if Earth is m₂, then m = m₂. We are also not interested in the negative solution of the equation since it's only indicative of direction.
So now we have the formula for escape velocity—and you could probably find this on Wikipedia, but I might as well do it right and deduce it.
So the gravitational constant: G = 6.674ᴇ-11, is the mass of the Sun: m₁ = 1.998ᴇ30 , and Earth's distance to the Sun: r₀ = 1.496ᴇ11.
So assuming we are moving the Earth in the direction of it's velocity—because we're smart—we can subtract the velocity the Earth already has from the desired velocity.
And we use the same equation for energy.
And because I'm too lazy to spit out any more equations I'm just going to let Wolfram do the rest of my work for me. Crunch, crunch, crunch...
The energy required would be around 4.56ᴇ32 Joules which is in the same neighborhood as the gravitational binding energy of the Earth which is 2.49ᴇ32.
