What is the speed of each particle relative to the other?

[pivoxa15]

Two particles are shot out simultaneously from a given point with velocity v. They start at right angles to each other. What is the speed of each particle relative to the other?

I got one particle thinks the other one is traveling at $$v\sqrt2$$ whereas they are traveling at v and vice versa. But I think I calculated it from a frame that is at rest relative to both particles and imagined what each particle would think if they had information in my frame.

 

[Andrew Mason]

Two particles are shot out simultaneously from a given point with velocity v. They start at right angles to each other. What is the speed of each particle relative to the other?

I got one particle thinks the other one is traveling at $$v\sqrt2$$ whereas they are traveling at v and vice versa. But I think I calculated it from a frame that is at rest relative to both particles and imagined what each particle would think if they had information in my frame.

If you stick to a vector analysis, it is not so confusing. The relative velocity of a to b is the velocity of a relative to the origin - velocity of b relative to the origin (or + the velocity of the origin relative to b). So just subtract the velocity vectors.

The relative velocity, therefore is $\vec v_a + (-\vec v_b)$. Relative speed is the length of that vector, which is the hypotenuse of a right triangle with sides of length v: $v_{rel}\sqrt{2v^2} = \sqrt{2}v$.

AM

 

[pivoxa15]

So veclocity of a to b is usually just $\vec v_a -\vec v_b$.

So if we look at another example, ship a is traveling at 4c/5. ship b is traveling at 3c/5 in 1D motion. The relative velocity of ship a to b or what ship b thinks a is traveling is $v_a -v_b = c/5$.

 

[jtbell]

If the speeds are relativistic and in different directions, you need to use the vector form of the relativistic "velocity addition" formula. See the section "The velocity addition formula for non-parallel velocities" on the following page from the Usenet Physics FAQ:

http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html

 

[pivoxa15]

But if I am just doing velocity addition for different frames (not events in frame), i.e know what the other frame is traveling relative to me, do I still need the relativistic addition formula? If I do apply it than I could pretend the 'event' is a person standing still in the other frame. So the velocity of that person is 0 in their frame S'. I get the speed I see them is v the relative velocity of the frames. This value can be calculated with the method I suggested in #3?

 

[Doc Al]

But if I am just doing velocity addition for different frames (not events in frame), i.e know what the other frame is traveling relative to me, do I still need the relativistic addition formula?

Yes. The relativistic addition formula tells you how velocities transform from one frame to another, which is what you want.