Time dilation (special relativity) problem

[palex3]

Homework Statement

This is a pretty classical problem. A plane goes from city A to city B. The distance between the two cities (measured on the ground) is 500 km. The plane is traveling at 0.2c. How long does the trip take for the pilot and what is the distance between the two cities for the pilot?

Homework Equations

I know that
$$L = L_0/\gamma$$
and
$$\Delta T = \gamma \Delta T_0$$

The Attempt at a Solution

The proper distance is 500 km, so I can work out that the pilot sees a distance of approximately 489.9 km. My problem is with the time. I know that I can divide 489.9 by 0.2c to get the time for the pilot: 0.00817 seconds. This is the correct answer. But why is it wrong for me to divide 500 km by 0.2 c (=0.00833s) and consider that the proper time, after which I transform it by multiplying by gamma (the result is 0.0085018 s)? I noticed that if I consider the time on the ground to be the "moving" time and the time for the pilot to be the proper time, using the formula gives me the correct time. But it doesn't make sense to me, since I started considering the ground as the rest frame. Why the switch? My guess would be that it has something to do with needing two clocks on the ground and one in the air, but I still don't really understand the asymetry.

Thanks for your help.

 

[alphysicist]

One common idea for proper time is that it is the time interval between two events in the reference frame at which they both occur at the same position coordinate.

Do you see how this means we can't set 0.00833 to be the proper time? (What are the x coordinates for the two events (plane is at A) and (plane is at B) in the ground reference frame? What are the x coordinates for the two events in the planes reference frame?)

 

[Doc Al]

My guess would be that it has something to do with needing two clocks on the ground and one in the air, but I still don't really understand the asymetry.

That's it exactly. The two events in this problem are: (1) Plane passes city A, and (2) Plane passes city B. On the plane, the time between those events is measured on a single clock. To the earth, that plane clock is moving and the "time dilation" formula can be applied. ("Moving clocks run slow.")

On the earth, the times of those events are measured on two different clocks. According to the plane observer, those Earth clocks are not synchronized (clock B is ahead of clock A), so the plane observer disagrees with the Earth observers as to how much time elapsed on Earth clocks during his trip.

But Distance = speed * time always works, as long as you stick to a single frame.

 

[palex3]

That clears it up, thanks a lot to both of you!
My problem was that I didn't see the difference between proper time (and lenght). Or rather, I didn't see what was special about it. In my head, time was time, no matter where you measure it (as long as it's in the same reference frame). Part of the cause for that missunderstanding is the naming convention in my physics book, part was me not being completely in the ''relativity'' mindset and part was my teacher using youtube videos to try to explain the issue.