How Do You Calculate Relative Speed in Special Relativity?

[Jalo]

Homework Statement

Two spaceships (A and B) are traveling in opposite ways. An observer in Earth measures both velocities.
The speed of ship A (Va) = 0.75c.
The speed of ship B (Vb) = 0.85c.
What's the speed of **** B regarding ship A?

Homework Equations

The Attempt at a Solution

At first I tried solving it using Galileu's relativity, but I've reached the result of 1.6c.
Then I remembered that when we're dealing with speeds close to c we have to use Einstein's special relativity, so I've looked around all the formulas I have but none of them involved velocity, only time, spaceship length, and there are also the ones involving energy but I don't think that has nothing to do with this exercise.

Thanks!

 

[Doc Al]

Read: Einstein Velocity Addition

 

[Jalo]

Read: Einstein Velocity Addition

I've used u'= -.85c and v=.75c

Substituting I got u= -.85c-.75c / 1-(-.85*.75) = -.977c

Therefore the speed of ship B regarding ship A is -0.977c

Is that solution correct? I think I've done everything right but i'd like to be sure :D

Thanks!

 

[Doc Al]

Good. The relative speed is correct, which is all you need. (The sign depends on your convention.)

 

[asteriatic]

Dear student,

You are correct in realizing that for speeds close to the speed of light, we need to use Einstein's special relativity instead of Galileo's relativity. In this case, we can use the relativistic velocity addition formula, which is given by:

V = (Va + Vb) / (1 + (Va * Vb)/c^2)

Using this formula, we can calculate the speed of ship B relative to ship A to be:

V = (0.75c + 0.85c) / (1 + (0.75c * 0.85c)/c^2) = 1.36c

This means that according to the observer on ship A, ship B is traveling at a speed of 1.36 times the speed of light. This result may seem counterintuitive, but it is a consequence of the relativistic effects of time dilation and length contraction. Special relativity allows for speeds greater than the speed of light to be calculated, but it does not violate the fundamental principle that the speed of light is the maximum speed in the universe.

I hope this helps clarify the solution to this problem. Good luck with your studies!