Ask the Physicist!

I have trouble accepting the idea that a spacecraft with a reaction rocket engine with a max exhaust velocity of say, 10,000kph, can somehow add up to give the ship a real velocity of say 40,000kph. (rocket engines can only have an exhaust velocity of about 10,000kph roughly speaking) The Relativistic reason given is that the engine is in a continuously renewing frame of reference that's effectively resetting itself to zero, then the rocket fires again or remains firing, and this adds the next 10,000 velocity to the last 10,000 kph velocity, making 20,000 kph realized speed. (relative to what?) The Relativists say that the rocket engine is only relative to itself, so its always just thinking that its got 10,000kph velocity, but somehow, using mystical magic, the rest of the spacecraft thinks that its relative to its origin, the Earth, so it CAN experience the additional speed effect. But aside from not making sense, this would mean that this re-zeroing of the ship's frame of reference, would enable a ship in space to reach incredible speed by continually combining any low-speed impulses. If this reasoning and scientific principal was correct then we can get to 40,000kph with a rocket engine that is only creating a gas velocity of say, 100kph. Also, the ship’s captain is calculating his speed by the frame of reference he set of from, ( the earth) but the poor engine thinks it's not moving so that apparently allows the rocket ship go faster than it's engines propelling gases? Has anyone got any thoughts on this? What practical experiment could be done on Earth in Earth atmosphere that could demonstrate this additive effect by re-zeroing one's frame of reference? We don't need to use rocket engines, or attain such speeds. Any demonstration of this effect at any speed will suffice. Don't say that we sent men to the Moon at 40,000kph as that is not explaining HOW this can work. I'm skeptical. I want some rational explanation and some way to replicate this principle, as if this is true, it seems to be a great way to hot up my car. A space ship is doing 40,000kph according to Earth, but if the guy in the ship can claim he is not moving (so his reaction rocket engine can give him an additive speed) then he can just as correctly claim he is actually already doing 500,000kph, and he will expect to get to Alpha Centauri ahead of schedule! Seems you can make up any frame of reference you want to suit your purpose! You can be travelling really fast, but claim that you are not, or be stationary but claim that you are moving! It makes no difference in the world of Relativists. But somehow it makes a real difference to the physics as measured from different reference frames. So next time I go to town in the bus, I want to say the bus is doing 1,500kph relative to some imagined frame of reference, as this will surely really be possible! An Earth based observer, watching the rocket, would measure it travelling at 40,000kph, but that would mean that the laws of Physics have been violated, as in the Earths reference frame, the rocket can’t go faster than its exhaust gas, which is only doing 10,000kph. However, at the same time, the spaceship observer would be saying he is not moving at all in relation to the engine, so it’s 10,000 velocity can compound up to 40,000kph because the gas doesn’t know it’s really already moving, maybe it can’t see the receding Earth? But also it is entirely possible that the ships observer CAN make his calculations from the Earth frame of reference, (that’s how he will know how far, what real speed, and in what direction he is travelling) so he will calculate a 40,000kph velocity again violating the laws of Physics. Please explain how this works.