# What is the speed of each particle relative to the other? (1 Viewer)

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#### 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 travelling at $$v\sqrt2$$ whereas they are travelling 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.

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#### Andrew Mason

Homework Helper
pivoxa15 said:
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 travelling at $$v\sqrt2$$ whereas they are travelling 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

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#### 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 travelling at 4c/5. ship b is travelling at 3c/5 in 1D motion. The relative velocity of ship a to b or what ship b thinks a is travelling is $v_a -v_b = c/5$.

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#### jtbell

Mentor
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

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#### pivoxa15

But if I am just doing velocity addition for different frames (not events in frame), i.e know what the other frame is travelling 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

Mentor
pivoxa15 said:
But if I am just doing velocity addition for different frames (not events in frame), i.e know what the other frame is travelling 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.

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