#### Orbital And Escape Velocities

In general, if you want a satellite with mass ms to orbit an object with mass m at radius r, you need to overcome the gravitational force Fg.

http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation

Fg=ms*m*G/r^2

with G=6.6*10^-11

You overcome this force by the Centripetal Force Fc from rotating at a radius r with speed v:

Fc=ms*v^/r

This means Fg = Fc

from which follows

ms*v^2/r = ms*m*G/r^2

which is

v^2 = m*G/r

This is the general formula as discovered by that educated Englishman Newton (who also tried to keep the Casino under control then, only to be poisoned by mercury). It is valid for any distance.

so if we plug in

mEarth = 5.9*10^24kg

and orbit 150 kms abobe the surface of the earth (r=6300km+150km) we get

v=sqrt(m*G/r)=sqrt(5.9*10^24kg*6.6*10^-11 Nm^2/kg^2/6450*10^3)

$ echo "scale=20;sqrt(5.9*10^24*6.6*10^-11/(6450000))" |bc

7769.94807082105368902512 (meters/s)

For 5000 kms we need

echo "scale=20;sqrt(5.9*10^24*6.6*10^-11/((6300+5000)*1000))" |bc

5870.27912378537946450721 (meters/s)

Now that's the orbital velocity, but not the launch velocity.

For that, calculate

LaunchEnergy = OrbitalEnergy, which is

LaunchEnergy = OrbitalKineticEnergy +OrbitalPotentialEnergy, which is

1/2*ms*vLaunch^2 = 1/2*ms*vOrbital^2 + INTEGRAL(h,h0,h1,Fg(h))

1/2*ms*vLaunch^2 = 1/2*ms*vOrbital^2 + INTEGRAL(h,h0,h1,ms*m*G/h^2)

INTEGRAL(h,h0,h1,ms*m*G/h^2) is

W(h1,h0) = ms*m*G/h0-ms*m*G/h1

so

1/2*ms*vLaunch^2 = 1/2*ms*vOrbital^2 + ms*m*G/h0-ms*m*G/h1

that is

vLaunch^2 = vOrbital^2 +2*m*G(1/h0-1/h1)

= m*G/h1 +2*m*G(1/h0-1/h1)

=m*G(2/h0-1/h1)

so

vLaunch = sqrt(m*G(2/h0-1/h1))

if we plug in the earths mass and a railgun supposed to lift something into 400 kms orbit we get:

echo "scale=20;sqrt(5.9*10^24*6.6*10^-11*(2/6300000-1/6700000))" |bc

8093.18507350298285516218

so we need

echo "scale=20;8093/340" |bc

23.80294117647058823529

Mach Numbers for that.

Btw. "Mach" stems from Ernst Mach:

http://de.wikipedia.org/wiki/Ernst_Mach