How can a rocket in space go several times faster than its own engine's gas velocity? Doesnt this break Newton's 3rd law? Gas velocity, 12,800kph, rocket speed 40,000kph?
The gas velocity is a constant 12,800 kph that accelerates a rocket from 0kph upto an arbitrary limit based on the quantity of propellant. As you burn propellant, your rocket velocity increases until you run out of fuel. If you increase your gas velocity your acceleration increases.
It's like saying 1kg of fuel in your car can only push you to 10mph, so how do cars travel faster than 10mph? They burn more fuel to increase their velocity.
The exhaust velocity is the velocity relative to your rocket.
That is, if we magically put a rocket into geostationary orbit, its velocity relative to the Earth is zero.
If we point the nozzle of that rocket at Earth, and ignore orbital mechanics beyond a couple kilometers around that rocket, the the exhaust gas will be heading towards Earth at 12,800 kph.
If, however, the rocket were well on its way to somewhere more interesting that some parking orbit around our planet, and was using Mars for a gravity assist to get further out of the solar system, we might expect the rocket to be moving past Mars at 40,000 kph.
Let's say that at periapsis -- the lowest point in our pass around a planet -- we turn our rocket back on, with the same exhaust speed of 12,800 kph. Someone on Mars would see our rocket accelerating "forward"... but would also see the exhaust gasses going forward as well, at 27,200 kph.
That's because in space, it doesn't matter how fast your rocket is traveling compared to something else, unless you're actively interacting with it.
Let's say your target is to land on Mars instead of just use Mars to get somewhere else... so instead of pointing your rocket so that the exhaust accelerates you prograde ("forward"), you do a retrograde burn, slowing your rocket down.
Now, at periapsis traveling 40,000 kph relative to your target on Mars, you fire your rocket, exhaust coming out the nozzle in front of you. Someone on Mars would see your rocket slowing down, and would see that initial burst of exhaust moving at an impressive 52,800 kph.
No Newtonian laws were broken here... The only thing broken is the assumption that there's some universal frame of reference for speeds that actually matter in space. There's only your local frame of reference that you should care about.
No this does not break Newton’s third law. The reason being that velocity is not conserved momentum is. If the mass of all the gas expelled from a rocket were the same as the mass of the rest of the rocket then the rocket would be limited to a speed of 12,800 kph (or whatever speed the gas was traveling at as exhaust velocities can vary depending on the chemicals used to produce them and their respective energies of the reaction).
But rockets tend to be flying fuel tanks with 90% or more of the mass of the rocket comprising propellant and only 10% rocket payload and structure. So after a 100 ton rocket has burnt 50 tons of propellant it still has 40 tons left (and 10 tons of payload) so it can reduce its mass by half again by burning 25 tons of propellant leaving 15 tons of propellant and 10 tons of payload and it can do that once more by burning 12.5 tons of propellant. Each of these steps would give the remaining rocket the same boost (all other things being equal).