The problem is that I am not exactly sure how I would use it (since there are not many examples in lua). From what i have understood, with this equation I can find the change in the velocity(?). Thanks in advance.

The delta v is how much you can accelerate a space craft (i.e. change its velocity) by expending its fuel in
e.g. a rocket motor.

You wouldn’t use it to simulate the kinematics of a craft, but you could use it to plan the different burns/maneuvers to perform to reach a given orbit, and then see if the craft has enough delta v for all of them.

You can implement it like this:

function craftDeltaV(exhaustVelocity, wetMass, dryMass)
return exhaustVelocity * math.log(dryMass/wetMass)
end

This assumes v_e is constant, which it usually isn’t. It changes with the atmospheric pressure from sea level to vacuum, and probably a bunch of other things.