Velocity Addition Inconsistancy

[BrettCollins]

I am interested in Special relativity and have noticed something i do not understand to do with velocity addition.If a spacecraft takes of from earth and is travelling at 3/4 the speed of light and launches a missile at a speed of 3/4 the speed of light in the same direction as the spacecraft. I used the relativity velocity addition formula W = (u + v)/(1 + uv/c^2) and this gives the speed of the missile as 24/25c (or 0.96c) if measured by an observer on earth. Now if the missile has a clock the special relativity time dilation equation say the the clock on the missile will be running at 16/25 slower. the clock on the spacecraft is running 4/5 slower to the observer on earth. An astronaut on the spacecraft will see the clock on the missile running 4/5 slower than the clock on the spacecraft. If the astronaut set their clock to run at the same speed as the clock on the missile and the observer on earth would then see the clock on the spacecraft as running (4/5)*(4/5) = 16/25 slower. I do not understand why this is not the same as observing the clock on the missile directly?Similarly the astronaut and the observer on earth would view the mass of the missile is different?

 

[swansont]

How are you calculating your dilation numbers?

[Strange]

I think you are using the wrong equation for time dilation. It should be:

[math]\Delta t' = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}[/math]

 

Which gives a time dilation of 0.6614378277661477 at 0.75c.

And 0.125 at 0.99215674164922c (the sum of 0.75 + 0.75)

[BrettCollins]

I am using:

t'/t = √((1- v^2/c^2 ))

Actually I did make a mistake when v = 3/4 (0.75) the ratio is √(7/16) but the two way to calculate time dilation of missile still come up with different values.

​to combine the time dilations you need to be multiplied, so the earth observer observing the astronauts clock that is synchronised with missile clock when the missile is moving at 0.75c is √(7/16) * √(7/16) = (7/16)

[studiot]

Brett, please note that Strange offered you a formula involving delta t or time differences.

 

Your formula involves t or time alone.

 

That implies some sort of absolute time or time synchronisation between two reference frames, which you cannot have.

Edited November 9, 2015 by studiot

[Janus]

I am interested in Special relativity and have noticed something i do not understand to do with velocity addition.If a spacecraft takes of from earth and is travelling at 3/4 the speed of light and launches a missile at a speed of 3/4 the speed of light in the same direction as the spacecraft. I used the relativity velocity addition formula W = (u + v)/(1 + uv/c^2) and this gives the speed of the missile as 24/25c (or 0.96c) if measured by an observer on earth. Now if the missile has a clock the special relativity time dilation equation say the the clock on the missile will be running at 16/25 slower. the clock on the spacecraft is running 4/5 slower to the observer on earth. An astronaut on the spacecraft will see the clock on the missile running 4/5 slower than the clock on the spacecraft. If the astronaut set their clock to run at the same speed as the clock on the missile and the observer on earth would then see the clock on the spacecraft as running (4/5)*(4/5) = 16/25 slower. I do not understand why this is not the same as observing the clock on the missile directly?Similarly the astronaut and the observer on earth would view the mass of the missile is different?

Because, as Studiot has already pointed out, time is relative and not absolute. This is demonstrated by the simple fact that time dilation is reciprocal. In your example above, the spaceship clock run at a rate of ~0.66 as measured by the Earth. Conversely, As measured from the spaceship, it is the Earth clock that is running slow by the same factor.

This means that as measured from the spaceship, The clocks on the Earth and missile run at the same rate. (while as measured from the Earth the missile's clock runs slow, and as measured from the missile, the Earth's clock runs slow.)

 

This may seem contradictory, but it really isn't. If you factor in length contraction and relativity of simultaneity into the problem, it all works out.( It has been my experience that most times when someone is confused about something in SR, it is because they are focusing too narrowly on one aspect of it, and excluding the other contributing factors.)