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.
You overcome this force by the Centripetal Force Fc from rotating at a radius r with speed v:
This means Fg = Fc
from which follows
ms*v^2/r = ms*m*G/r^2
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
$ echo "scale=20;sqrt(5.9*10^24*6.6*10^-11/(6450000))" |bc
For 5000 kms we need
echo "scale=20;sqrt(5.9*10^24*6.6*10^-11/((6300+5000)*1000))" |bc
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)
W(h1,h0) = ms*m*G/h0-ms*m*G/h1
1/2*ms*vLaunch^2 = 1/2*ms*vOrbital^2 + ms*m*G/h0-ms*m*G/h1
vLaunch^2 = vOrbital^2 +2*m*G(1/h0-1/h1)
= m*G/h1 +2*m*G(1/h0-1/h1)
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
so we need
echo "scale=20;8093/340" |bc
Mach Numbers for that.
Btw. "Mach" stems from Ernst Mach: