What Happens When Things Go Near the Speed of Light?

Advanced Analysis: Deriving the formula for g

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Use the mathematical form of the plot to derive g.

Even though the observations disagree, there is a way to reconcile all of this: the hyperbola.
Remember the equation for our hyperbola is time2 - space2 = t2

If you stand still watching the meson travel through the detector, then:

Ty2 - Sy2 = t2 = Tm2

where your time = Ty your space = Sy
  meson's time = Tm meson's velocity = Vm
But your distance is how far the meson travels before it decays. This distance is the same as the meson's velocity multiplied by the time that you measure for its decay.

Ty2 - (VmTy)2 = Tm2

Ty2 - (Vm2Ty2) = Tm2

Ty2 (1 - Vm2) = Tm2

Ty (1 - Vm2)1/2= Tm

This hyperbola still uses the same unit for distance as it does for time. Remember that you made velocity unitless so that you could express it as a fraction of the speed of light.

Take a look at the last equation. To find the time in the meson's frame, you simply take the time as you observed it in your frame and multiply it by a correction factor. Physicists call this correction factor gamma (g).

g = (1 - Vr2)-1/2

Ty/g = To

where other observer's time = To
and relative velocity = Vr