Finding the maximum and minimum velocity A $K^0$ meson travels at a velocity of $0.9 c$ when it decays into a meson $\pi^{+}$ and into a meson $\pi^{-}$. What are the maximum and minimum speed that the mesons can have?
I'm considering two reference systems, that of $ K^0 $ and that of an observer who sees $ K^0$ moving at a velocity of $0.9c$ and hence trying to transform the velocities, but it is not clear how to proceed or if this is correct. Any hint?
 A: Just a couple of hints. Firstly, in the CM frame the $K$ meson has no velocity, so it's four momentum if $p=(m_K,0,0,0)$. Try to use in an intelligent way the conservation of momentum to find the three momenta of the pions. Remember that in the CM frame the pions are being ejected in a very special way.
Moreover, remember that the boost factor $\beta$ can be found from the three momenta and the energy of a particle by means of $$\beta = \frac{|\mathbf{p}|}{E}$$ and that, in the SI system, $v=c\beta$.
This is all you need, plus some Lorentz transformations that you can easily find with the other useful relation $$\gamma = \frac{E}{m}$$
A: Here's a geometric method that should give the same result as @DavideMorgante's approach.
You can proceed as you would a Euclidean problem...
but you have to use hyperbolic trig functions [of the rapidity],
as well appreciate that the hyperbola plays the role of the circle.
(Rapidities are the Minkowski-angles between [future] timelike-vectors. cos[h]=adjacent/hypotenuse , tan[h]=opposite/adjacent, etc...).
Draw an energy-momentum diagram of the decay.
By conservation of 4-momentum, you'll get a triangle whose sides you know (since you know the [rest] masses).
For simplicity, first consider the case in the CM-frame [where the long side is vertical].
You can figure out the [rapidity] angles made with the long side. (Use components or the law of cosines (dot products) [in Minkowski spacetime].)
From each [rapidity] angle, you can get the [velocity] slope [with respect to the vertical].
For the lab frame, you have to [boost] rotate the triangle by the appropriate [rapidity] angle.
Find each [rapidity] angle that the sides make with the vertical [lab frame worldline].
From each [rapidity] angle, determine the corresponding  [velocity] slope.
This can also be formulated with a calculation with 4-vectors and dot-products.
However, I feel the geometric method will guide your intuition on how to proceed on this and on other problems.
