And additionally, would it ever be possible to launch something into deep space and get entirely left in the dust by the solar system whizzing by? Completely untethered from its gravity?
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First of all, you can’t just say “200km/s”. This is because **speed is relative**.
If I say my car is going 80km/h, that means something to you only because we are both on the surface of the earth. The vast majority of the time, we are observing and measuring the speed of things around us relative to the ground. We are so used to having big and seemingly unmoving things around us, that we don’t even think about the fact that we are using them as a reference.
In space, there’s no ground. I know that sounds obvious, but when you think about what speed means when there’s nothing to move relative to, it will be a lot less obvious. Imagine a very lonely planet, that’s the only planet in the whole world. If you looked up at the sky from that planet, even with the best satellites, you would see absolutely nothing. Pitch black all around you. Now: how fast is the planet going?
What does that even mean? You can’t have a speed if you’re the only thing. It has to be you and another thing. Then you can ask “what is my speed relative to that thing?” So before you measure the speed of an object, you have to choose another object as well.
For example, if you tell me the speed of the planets in the solar system is 200km/s, I can whip out my speedometer, point it at the ground, and measure the speed of Earth to be zero (relative to myself, because I’m holding the speedometer). Then I could point to the speedometer and tell you that you’re wrong. But neither of us is wrong, we just chose different reference frames.
The 200km/s number is most likely calculated between the centre of our solar system (which is the sun) and the centre of the milky way galaxy (which happens to be a supermassive black hole called “Sagittarius A*” or Sgr A* for short). The planets also have their own (orbital) speed relative to the sun, so their speed relative to Sgr A* would be different from the sun’s and also periodically changing. Imagine mounting a speedometer to the trunk of a truck, then measuring the speed of cars passing by while doing donuts with the truck. Sometimes the speeds match up and add, sometimes the speeds partially cancel out, sometimes neither.
Now that that’s hopefully clear, we can think about that rocket. Before it launches, its speed is 0 relative to the ground. Relative to the sun, its speed is around 30km/s. Relative to Srg A*, its speed is roughly 200km/s, give or take 30. Relative to someone driving on the highway, its speed is around 100km/h.
You might be tempted to wonder “which of those speeds is the real speed?” but try as hard as you can to really internalise the fact that there is no such thing as a single speed that is the only real one.
You might also notice that I didn’t mention its speed relative to its destination. That’s because with everything like the spin of the Earth around itself and the planets orbiting the sun at different speeds, the speed of the rocket relative to its destination is always changing. Because we don’t want to waste fuel, we usually launch it near the equator and try to time the launch so that the rocket is already heading towards its destination before it launches, basically using all of those spins and orbits as a slingshot. That way, once it climbs out of Earth’s gravity, it doesn’t have to “swim against the current”, so to speak. Of course its destination is not going to stay still while the rocket is on its way, so things are complicated. Not to mention gravity assists and everything else. It’s rocket science after all.
And yes, before during and after the rocket makes its journey from Earth to whatever destination it has (as long as the destination is also orbiting the Sun) its speed relative to Sgr A* stays roughly within that 200km/s range. Yes the rocket speeds up and slows down (acceleration, unlike speed, is not relative) but a kilometre per second is such a crazy large unit of speed that the amound of fuel the rocket has to accelerate is nothing but a rounding error in 200km/s.
And finally for your last question: regardless of the gravity of the solar system, a rocket that launches from the solar system would have around the same 200km/s speed relative to Sgr A*, and it would be orbiting around Sgr A* along with the solar system. The only way to stop doing that is if it was a magical rocket that never ran out of fuel; it could accelerate in the opposite direction until its speed relative to Sgr A* was zero. Then, its speed relative to the solar system would be around 200km/s.
But would that look like being left in the dust by the solar system, or would it look like leaving the solar system in the dust? Is there a difference?
Imagine a helicopter on the ground at the equator. The speed of the ground relative to the centre of Earth is around 460m/s. Before the helicopter takes off, its speed relative to the ground is 0. That means it is also moving at 460m/s relative to the centre of Earth. If the helicopter takes off and just hovers, its speed will be the same from both reference frames. If the helicopter tries to reduce its speed relative to the centre of Earth, it will not look like it’s slowing down. It will not look like the ground is moving away from the helicopter. It will just look like the helicopter is accelerating west.
This is again because we are so used to using the ground as a reference frame. If I crash my car into a wall and tell everyone that the wall was coming towards me really fast, people would laugh at me. “You moved towards the wall, not the other way around!” they would say. But in space, there’s no ground. **There is no difference between the rocket “whizzing by” the solar system and a solar system “whizzing by” the rocket**.
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