Accelerating Twice: Astronaut Achieves .99999c in Nuclear Ship

[Buckethead]

An astronuat in a nuclear ship accelerates away from Earth and gets very near the speed of light, .99999% in fact. The accelerometer on the ship indicates the ships acceleration has been decreasing during this feat as the mass of the ship has increased greatly and the nuclear engine can no longer continue to accelerate the craft to any great degree. The engines are cut, the astronaut is happy that he has achieved the fastest speed ever, and after 10 years of coasting at this speed he calculates that he has just traveled past all galaxies and into a deep part of space where no stars exist.

He dies, but a stowaway baby boy was on board when the ship left Earth and is now 10 years old. The boy looks at the accelerometer and it reads 0. There is no speedometer as speed is relative so they didn't put one on the sip. The boy figures the ship is at rest and fires up the engines, but to his surprise the ship does not move (accelerate) more then just a tiny tiny amount as the ship is already traveling so fast and further acceleration is almost impossible. The boy knows the thrust of the engine, knows the rest mass of the ship and can read the accelerometer and from this was able to determine how much mass the ship now has. From this he was able to determine the ship is traveling at .99999c or in other words the absolute velocity of the ship relative to nothing. Where is the flaw in my logic?

 

[Doc Al]

Your flaw is in assuming that the ship's mass increases from the view of the ship.

 

[JesseM]

An astronuat in a nuclear ship accelerates away from Earth and gets very near the speed of light, .99999% in fact. The accelerometer on the ship indicates the ships acceleration has been decreasing during this feat as the mass of the ship has increased greatly and the nuclear engine can no longer continue to accelerate the craft to any great degree.

That part doesn't make sense--the mass of the ship doesn't increase in its own rest frame, so if the force from the nuclear engine is constant in the engine's own rest frame, there'd be no reason why the acceleration as measured by an accelerometer on the ship would decrease. A ship can in theory accelerate at constant acceleration in its own frame forever (though the fuel requirements would quickly become unwieldy), though as viewed in the frame of some inertial observer outside the ship, the acceleration will be constantly decreasing as it approaches light speed in that frame (and hopefully you understand that all measurements of speed are relative, we can only say the ship is moving at 0.99999c in the a given frame, there isn't any objective truth about how fast it's going).

He dies, but a stowaway baby boy was on board when the ship left Earth and is now 10 years old. The boy looks at the accelerometer and it reads 0. There is no speedometer as speed is relative so they didn't put one on the sip. The boy figures the ship is at rest and fires up the engines, but to his surprise the ship does not move (accelerate) more then just a tiny tiny amount as the ship is already traveling so fast and further acceleration is almost impossible.

As I mentioned above, this just isn't how it works. You are free to accelerate at 1G forever as measured by an accelerometer on the ship, but you'll still never reach light speed in the frame of an external inertial observer, and in their frame your acceleration will be constantly decreasing. See http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html for more details.

 

[Buckethead]

Your flaw is in assuming that the ship's mass increases from the view of the ship.

I thought that was how it worked. OK, so from the ships viewpoint the mass has not increased. Does this mean when the engines are fires up, the ship will show a large acceleration? But if the ship was already moving at .99999c, then how could it show a new rapid acceleration which would easily take it past c?

 

[JesseM]

I thought that was how it worked. OK, so from the ships viewpoint the mass has not increased. Does this mean when the engines are fires up, the ship will show a large acceleration? But if the ship was already moving at .99999c, then how could it show a new rapid acceleration which would easily take it past c?

Because the acceleration in the ship's own current rest frame is not the same as the acceleration in the frame where the ship is moving at 0.99999c. Each frame uses their own rulers and clocks to measure distance and time, and each frame sees the other frames' rulers as being shrunk relative to their own, and the other frames' clocks to be slowed-down and out-of-sync.

 

[DaveC426913]

While the engines are firing, the craft will accelerate at 1g from the pilot's PoV. He could do this forever if he chose to.

But from the PoV of someone on a planet, his v will approach c closer and closer but never quite reach it. The pilot inside (if he could see him) will appear to be virtually frozen in time.

 

[Buckethead]

Thanks for clearing some of these things up. So if the pilot is maintaining a 1g force according to his accelerometer and he does a fly by past Earth at near the speed of light and someone on Earth snaps a picture of the accelerometer, they will see a 1g reading on the instrument, yet by Earth based measurements the ship does not seem to be accelerating. How can this be?

 

[JesseM]

Thanks for clearing some of these things up. So if the pilot is maintaining a 1g force according to his accelerometer and he does a fly by past Earth at near the speed of light and someone on Earth snaps a picture of the accelerometer, they will see a 1g reading on the instrument, yet by Earth based measurements the ship does not seem to be accelerating. How can this be?

Again, each one is using their own rulers and clocks to measure time and distance, and of course each one defines velocity as change in measured distance over change in measured time, and defines acceleration as change in measured velocity over measured time. The rulers and clocks of different observers do not agree with each other.

 

[DaveC426913]

Thanks for clearing some of these things up. So if the pilot is maintaining a 1g force according to his accelerometer and he does a fly by past Earth at near the speed of light and someone on Earth snaps a picture of the accelerometer, they will see a 1g reading on the instrument, yet by Earth based measurements the ship does not seem to be accelerating. How can this be?

Think of how the two observers experience time. The Earth-bound observer with his telescope sees a clock aboard the spaceship. The clock ticks off one second every ten years from Earth's PoV. And ten years is how long it took the spaceship to accelerate from speed a to speed a+g (i.e at 1g acc.)

But aboard the ship, that same speed change only took one second.

 

[LURCH]

OR...

Thanks for clearing some of these things up. So if the pilot is maintaining a 1g force according to his accelerometer and he does a fly by past Earth at near the speed of light and someone on Earth snaps a picture of the accelerometer, they will see a 1g reading on the instrument, yet by Earth based measurements the ship does not seem to be accelerating. How can this be?

The accelerometer is a small weight on the end of the spring. Consider that the mass of the weight inside the accelerometer has increased by the same proportion as the mass of everything else aboard the ship. To the man on board the ship, the 1 kg weight is being subjected to 1 G of acceleration, and so it shows a reading of 1 G. To an observer on the ground, the ship's mass has increased by a multiple of five, so that the engines are only able to accelerate the ship at a rate that we would calculate to generate 1/5 G.

But then, the little weight on the end of the spring inside the accelerometer has also become five times more massive. So, the earthbound observer sees a 5 kg weight subjected to 1/5 G of acceleration, resulting in a reading of...1 G.

For anyone else who mgith be reading along, these last three posts may seem like they're giving three different explanations, but they are all actually different ways of saying the same thing.