Question

Could a coil gun's projectiles escape earth's gravity?

Answer 1

Acceleration due to gravity as a function of distance from the center of the earth r, constant G, and mass of earth M, where the positive vector direction is away from the center of the earth, thanks to Newton:\[\vec a_g = -\dfrac{GM}{r^2}\hat r\]Integrate to evaluate the gravitational potential between the points r=R (radius of earth, and thus the surface of the earth) and to infinity (take note that the second negative in the second integral appears due to the dot product of two opposing vectors):\[\phi_g=\int_R^\infty \vec a_g \cdot d\vec r = -\int_R^\infty -\dfrac{GM}{r^2}dr = \left.-\dfrac{GM}{r}\right|_R^\infty = \dfrac{GM}{R} - \lim_{r\rightarrow \infty} \dfrac{GM}{r} = \dfrac{GM}{R}\]Kinetic energy per mass is as follows:\[\dfrac{E_k}{m} = \dfrac{v^2}{2}\]The minimum velocity for a projectile to be launched to escape would satisfy the equality between the gravitational potential difference between the surface of the earth and a distance infinitely far from the earth, and the kinetic energy per mass.\[\phi_g=\dfrac{E_k}{m}\]\[\dfrac{GM}{R}=\dfrac{v^2}{2}\]\[v=\sqrt{\dfrac{2GM}{r}}\]Since this came out to be a definite value, there is thus a theoretical escape velocity for projectiles shot from the surface of the earth. This is, of course, not counting for air friction.

Answer 2

Coil guns are much more less developed than linear barrels, and they are proyected to have a range of 300 nautical miles for ships turrets. Coil guns should be able to shoot a proyectile out of earth gravity but the gun itself as far a current technology goes would be gargantuan. With better electronics, new capacitors, new composites, alloys and some whisperer they can overcome the barrier.

Answer 3

If you're a whisperer, you can.