# Relative motion

1. Nov 30, 2005

### moose

I know that you are all sick of these threads, but....

If you have object A and B close to each other
A-B

Then B accelerates to some high velocity, for example 3/4 of the speed of light in a matter of few seconds, so extremely good acceleration. Then, at this speed it keeps moving along at 3/4 c (relative to object A) for a long trip, until it reaches 3/4 light years away from object A. Now, object B accelerates back to object A at an extremely quick acceleration and then cruises at a high speed, until it reaches object A when it stops.

If clocks were placed on both objects, why would object B show that less time has gone by then object A? I understand why time changes, but why does A seem to have gone through more time if the ONLY difference in the two objects is that object B accelerated a few times, other than that, it could have thought that object A was going away from it. So this goes to my next question, if something is traveling at a constant velocity, does time change more and more as the object moves for a greater distance?

I'm asking my first question because when they both have constant velocities, not accelerating whatsoever, both feel stationary.

2. Nov 30, 2005

### JesseM

No. No matter which inertial frame you use to analyze this problem, both clocks will be ticking at a constant rate if they are moving at a constant velocity. You'd be free to analyze this problem from a frame where A and B are initially moving at 3/4c, then B comes to rest while A continues to move at 3/4c, then B accelerates to some velocity even higher than 3/4c to catch up with A. From the point of view of this frame, during the first leg of the trip when B was at rest its clock was ticking faster than A, but then during the second leg when it was moving at a greater velocity than A its clock was ticking slower, and then end result will be that this frame will still predict that A's clock will be behind B's when they reunite. No matter which inertial frame you use to analyze the problem, you will always get the same answer to the question of what the two clocks read when they reunite, even though you get different answers to how fast each clock was ticking during the two legs of the trip.

You can think of it in a geometric way...if you draw two points on a piece of paper, and draw two paths between them, one which is just a straight line between the points and one which has a bend in it, then the straight line will always be the shortest distance between the two points. Similarly, the geometry of spacetime is such that if you have two paths/worldlines between two events, one which is straight (no accelerations) and one which has a bend (the turnaround), then the straight path will always have the greatest proper time (time as measured by a clock that travels along that path).

3. Dec 1, 2005

### ___

well, i dont want to hijack the thread but...
lets say A and B are stationary in relation to each other. now A travels as fast as possible away from B. after a year of A (and lets say 10 yrs. of B.{i cant do the calculation}), A turns in opposite direction and travells at same speed for same time(1 yr for A and 10 for B).
now here is the question.
will the time of A & B match?
isn't velocity vector and doesn't travelling at one direction and coming back equal to 0 velocity.
so wont the time of A and B match exactly?
thanks.

4. Dec 1, 2005

### JesseM

No, you can't calculate total time dilation by taking the average velocity and plugging that into the time dilation formula, it just doesn't work that way. To find the total time elapsed on A's path, you must add the time elapsed on each constant-velocity portion of the trip, in this case 1 year + 1 year = 2 years. And to find the time elapsed on a curved path where the acceleration is not instantaneous, you'd do the calculus trick of approximating the curved path by a bunch of constant-velocity segments and then looking at what happens as the size of the segments approaches zero.

5. Dec 1, 2005

### ___

thanks. but i m still confused.
as u said that there will be a curve and there is non-instantaneous acceleration, i would like to modify the experiment.
lets say A went away from B at 100kmph. it took it 1 minute to reach that speed from 0. after a year and a minute(time for its acceleration) it used its propellor at the front for a minute to reach 0 speed(which has same power as that of the first propeller). what i mean is that there was no curve (or maybe a world with only one satial dimention, but i think it will only serve to complicate the experiment).
and then 1 minute for acceleration in opposite direction and a year and a minte of retardation. what would be the case here?
thanks again.
:)

6. Dec 1, 2005

### JesseM

I didn't mean a curved path through space, I meant a curved path through spacetime--any acceleration will cause such a curved path. Imagine a universe with only one spatial dimension, and draw the x-axis representing space, and then at right angles to that draw the t-axis representing time. Now if you draw the path of any object moving at constant velocity, you can see it's a straight line; but if you draw the path of an object that's accelerating, you can see it will look like a curve in this diagram.

7. Dec 1, 2005

### ___

ok, i get it. thanks