# What happens when you apply a force in space over infinite time?

1. Aug 26, 2008

### danben

Say you're in a spaceship, or someplace where there is no gravity, friction or resistance, and you have a rocket engine with a LOT fuel. You power up the rocket engine and it delivers consistant thrust for near enough infinite time. What exactly happens?
As far as i understand you would go faster and faster, until you reach close to the speed of light... but that's where things get confusing for me. I know you can't go faster or really approach the speed of light, because it requires infinite energy to accelerate mass right?
So what happens in the above? Is there some sort of extra resistance that's created at those speeds?

From what i can tell, the increase in speed from the constant rocket engine would be linear, so it'd only be a matter of time before you reached light speed. Or is there some trickery with time at those speeds?

2. Aug 26, 2008

### Staff: Mentor

Due to the theory of Relativity, you would feel an acceleration force all the time, but if you look out your window, you find your speed never surpasses the speed of light. The faster you go, the worse that linear approximation gets.

3. Aug 26, 2008

### HallsofIvy

Staff Emeritus
In a sense yes, there is "extra resistance". F= ma applies only for constant "mass". Since the relativistic mass increases with speed, for fixed force the acceleration decreases with increasing mass. The longer you run the rocket engine, the closer to the speed of light you get but it is an "asymptotic" approach not linear.

Last edited by a moderator: Aug 29, 2008
4. Aug 28, 2008

### atyy

5. Aug 28, 2008

### Staff: Mentor

Last edited by a moderator: May 3, 2017
6. Aug 28, 2008

### MeJennifer

This is simply untrue and a common recurring error. A rocket with a constant thrust does not need increasingly more force to maintain a constant acceleration. If anything it needs less as the burned up fuel will lower its rest mass.

7. Aug 28, 2008

### GRB 080319B

Doesn't relativistic mass increase faster than the decrease in the rest mass of the spaceship by the burning of it's fuel as the ship approaches c? If so, does relativistic mass increase exponentially as the spaceship approaches c? Is there a relationship between the increase in relativistic mass of the ship and the decrease of the rest mass of the ship as it burns fuel? Does faster acceleration equate to faster increase in relativistic mass? Is the inertia of the relativistic mass what causes the ship to resist accelerating to the speed of light? Any help or corrections will be appreciated. Thank you.

8. Aug 28, 2008

### granpa

the rocket itself wouldnt notice the increase in relativistic mass. everything would seem normal to a person onboard the rocket.

9. Aug 29, 2008

### jacksnap

so what would show on the speedometer of such a spaceship,
would you just approach 299 792 458 m / s and sail past it without noticing?

10. Aug 29, 2008

### atyy

Yes, MeJennifer is probably correct, because in the instantaneous inertial frame of the rocket, the force and acceleration are constant by definition, and accordingly, the rate at which resources are used too (at least by intuition, I didn't check this).

11. Aug 29, 2008

### DrGreg

That depends on how the speedometer works.

One way would would use the radar principle: the ship sends a radio signal to Earth, the signal is echoed back immediately, and the ship then calculates the distance from Earth from the speed of light, divides by ship time to get a velocity. By that method the speedometer would never quite reach 299 792 458 m/s. This is what we mean by "velocity" in relativity.

Another method: imagine there are already "milestone" buoys in position, stationary relative to Earth, each marked with the distance from earth measured in the Earth's frame. The ship uses these buoys to measure distance and divides by ship time to get a velocity. Length contraction means that the first method measures a shorter distance than the second method. This second method measures a quantity called "celerity" (also known as "proper velocity"). The celerity is always greater than velocity, although at low speeds they are approximately the same. It turns out that this method gives an ever-increasing, limitless celerity. The celerity of light is infinite.

12. Aug 29, 2008

### atyy

One thing I am not clear about is whether the "relativistic rocket" (constant force and acceleration in a constantly changing reference frame) is equivalent to HallsofIvy's scenario (constant force in the original reference frame). If they are different, that may sort out apparent contradictions.

Anyway, the heuristic for seeing that you cannot accelerate to the speed of light, no matter what you do, is:
1) Relativistic mass approaches infinity as the speed of light is approached.
2)E=mc2, so the energy required is infinite, even though you are only trying to reach a finite speed.

Last edited: Aug 29, 2008
13. Aug 29, 2008

### DrGreg

It turns out, surprisingly, that, for linear motion of a constant rest-mass, the force is invariant, i.e. all inertial observers agree on its value.

So constant proper acceleration (measured in constantly changing co-moving inertial frames) means constant force (in any frame) on a constant rest-mass, but does not mean constant coordinate acceleration measured in any single inertial frame.

For the technical details, see this post of mine. That post also has links to the equations of motion parameterised by the rocket's own proper time, and my post #22 of the same thread gives the equations in inertial coordinate time (all assuming constant proper acceleration). (And, for what it's worth, celerity is $dx/d\tau = \gamma v = c \sinh \phi = p / m$, in that post's notation.)

Note, however, that a rocket burning fuel does not have constant rest-mass, so this result doesn't hold in that case.

14. Aug 31, 2008

### yuiop

What you say is true if we are talking about proper acceleration.

If we are talking about acceleration as measured by an external inertial observer then it would require progressively more and more fuel per second to maintain constant acceleration relative to the inertial observer.

15. Sep 7, 2008

### MeJennifer

It is very simple, if the rocket applies a constant thrust then the fuel consumption does not rise if on the other hand a rocket applies an increasing thrust then obviously fuel consumption increases as well.