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I'm attempting this problem:

Rocket undergoes constant acceleration from rest until it reaches a speed of 0.9c. The astronaut in the rocket feels a constant "induced gravity" of 1g. How long does it take the rocket to reach constant velocity in the inertial frame? How about from the reference frame of the rocket?

For the inertial reference frame I did this:

$$v(t)={a_0t\over {\sqrt{1-{v^2\over c^2}}}}$$

$a_0$ is the proper acceleration, which is $9.8 m/s^2$, and $v=0.9c=0.9 \times(3 \times10^8 m/s)$

$$0.9 \times(3 \times10^8 m/s)={9.8m/s^2t\over {\sqrt{1-{0.9^2}}}}$$ $$t=1.2\times10^7s=0.38yr$$

To calculate how much time it took from the reference frame of the rocket I did:

$$d\tau=dt\sqrt{1-{v^2\over c^2}}=0.38yr\sqrt{1-0.9^2}=0.1649yr$$

Where $d\tau$ is the proper time.

I know this is wrong because I used an online calculator (here), and it says that the intertial frame (Earth) says it takes $2$ years, not $0.38$ years. And that the rocket says it takes $1.4262$ years, not $0.1649$ years.

What am I doing wrong?

Edit: The first equation I found in my textbook for motion under a constant force. I assumed that the force would be constant because it's an induced gravitational force that is said to be constant. I set $v(t)$ equal to $0.9c$ because that's the speed it will be at time $t$ which is what I'm looking for.

I know that I can't use the equations for constant velocity because here the velocity is not constant, the acceleration is. I think it may not be working for me because $v(t)$ is not equal to $v$. However, I'm not sure how to find $v$ here because it varies.

I visited the Wikipedia page for Space travel using constant acceleration, but it was difficult to follow.

Is this not working for me because the $v$ isn't constant?

Edit 2: The Wikipedia page was difficult for me to follow because it involved hyperbolics. We haven't talked about hyperbolic motion in detail in my class, it was only mentioned once, very briefly. So I thought that there could be a way to do this problem without using $\cosh$, $\sinh$ and $\tanh$

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marked as duplicate by M. Enns, user191954, Aaron Stevens, John Rennie homework-and-exercises Dec 2 '18 at 9:53

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    $\begingroup$ I don't understand. The rocket is accelerating at constant acceleration $a_0$ and to find time dilation you applied the formula for constant speeds? I don't think you can do that. $\endgroup$ – João Vítor G. Lima Dec 2 '18 at 4:54
  • $\begingroup$ You need to boil this down to a conceptual question for this to be on-topic for the site. For example you might explain where you got your first equation from and work through why it is wrong and ask for help wherever you get stuck. $\endgroup$ – CR Drost Dec 2 '18 at 4:54
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    $\begingroup$ @João Yeah, the proper expression involves a hyperbolic tangent... $\endgroup$ – CR Drost Dec 2 '18 at 4:56
  • $\begingroup$ @CRDrost I apologize, I've been working on this problem for a long time. I added an edit to hopefully clarify my question. $\endgroup$ – matryoshka Dec 2 '18 at 5:07