Relative Velocity in Special Relativity?

Question

In an inertial frame two particles are shot out simultaneously from a given point, with equal speeds v, in orthogonal directions. What is the speed of each particle relative to the other?

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LMC
In an inertial frame two particles are shot out simultaneously from a given point, with equal speeds v, in orthogonal directions. What is the speed of each particle relative to the other?
The answer is given as v = u(2-u^2/c^2)^0.5

Answer 1

OK... here goes me desperately trying to remember things I've not thought about for years...

Since both particles are in the same inertial frame, it's not really a relativistic problem until the speeds are approaching the speed of light. So treat it as classical mechanics.

If the velocities, v, are orthogonal (remember they are not speeds, they are velocities) that means they are 90 degrees to each other (doesn't it? I think so but I could be wrong).

If we say particle A is going along the x axis, and particle B along the y axis, we have...

v(A) = (v, 0) and v(B) = (0, v)

To relative velocity is going to be:

v(A) - v(B) = (v, 0) - (0, v) = (v, -v)

To work this out as a speed is simply Pythagoras, and gives us an answer of... v!

This means that if you launch two particles at the same speed, v, at right angles to each other, their relative speed is just the same as the speed they were launched at! Wow!

Thing about it, though, and this makes sense... If you launched them in the same direction, their relative speed would be zero (they'd be travelling side by side and could happily wave at each other - hello!). Launch them in opposite directions and the relative speed would be 2v. So... 90 degrees is halfway between the same direction (0 degrees) and the opposite (180 degrees), so that's halfway between 0 and 2v... which is v.

Now, if you want to use speeds close to the speed of light, you'll have to use a factor of the constant 'gamma' from the Lorentz transform equations, which depends on how close v is to the speed of light. It's messy at best... too messy for a Yahoo answer!

Hope this helped and didn't confuse you too much!

Answer 2

The relative velocity is the vector sum of the two velocities.In relativity, the Relative velocity is by definition a vector sum.

Hence, if two moving masses move toward each other ,the relative velocity is their approach velocity. However The collison velocity is equal to the square root of the sum of the square of the two velocity which is always less in magnetude than their velocity of approach.

When two masses move inertially parallel to each other at the same speed,Their relative velocity is zero ,as Per the Principle of Relativity that Galileo Postulated.

Answer 3

Heck if I know the right answer... But I would guess that it depends on if the space the two particles occupy is symmetric...in which case the distances between both particles would be change depending on how 'wiggly' the space is... making their velocities varied.

But if the space is flat or at least symmentric then the relative speed of each particle to each other would be zero... in other words they wouldn't appear to move relative to each other...

I think, but don't quote me on that...

I need to do some more reading... Great question though!