Observing from a reference frame moving with high speed

Question

Suppose at a distance $x$ from earths surface,a rocket comes with $v$ velocity which is comparable with the speed of light. Suppose lifetime of the rocket is $T$. We want to know whether the rocket will land or earth or not.

Here is my approach when observing from earth frame.

$T$ is the proper time interval and the time we will be counting is $t$ which is the improper time. So by time dilation formula,we will have $t$ and by $s=vt$ we can easily deduce the distance the rocket travelled and reason.

But the books did something else.

They observed from the frame of the rocket and they say that due to length contraction,the $x$ distance that was there earlier will seem to be $x\gamma$. But i don't understand how are they saying it. First of all proper length is something which is defined when the length is at rest with respect to the frame. When they observe from rocket frame, how is the distance from earth to the rocket a proper length? How is $x$ in the frame of reference of the earth?

The approach I came up with is really intuitive but i don't understand the solution the books came up with. Maybe i haven't fully grasped the concept. Clearing my misconceptions will be really appreciable.

Answer 1

Both your approach and the book's approach are fine. There's no absolute motion, only relative motion, so we can either take the Earth to be at rest, or the rocket to be at rest. Your approach takes the Earth to be at rest; the book's approach takes the rocket to be at rest and the Earth to be moving toward it. $x$ is distance between Earth and where the rocket starts, as measured by Earth. In the rocket's frame this distance will be length contracted, since Earth (and the starting point at rest with respect to the Earth) are moving past the rocket at speed $v$.