Lambda \(\left(\Lambda^{0}ight)\) baryons can be created in high-energy \((p, \bar{p})\) collisions of protons and antiprotons in the reaction \(p+\bar{p} ightarrow \Lambda^{0}+k^{+}+\bar{p}\), where \(k^{+}\)is a positive \(k\) meson.

(a) Find the minimum (i.e., threshold) energy required for the incident antiproton if the target proton is at rest in the lab. The masses of the particles (in \(\mathrm{MeV} / \mathrm{c}^{2}\) ) are \(p\) or \(\bar{p}: 938.3 ; \Lambda^{0}: 1115.7\); and \(k^{+}: 493.7\).

(b) Find the minimum energy of each initial particle in a collider experiment, in which the total momentum is zero.

(c) Suppose that in the collider experiment the energy of each initial particle is twice the minimum energy required. Find then how far the subsequent \(\Lambda^{0}\) will travel in the collider detector before it decays, assuming the \(\Lambda^{0}\) lasts for a time \(2.63 \times 10^{-10} \mathrm{~s}\) (the mean lifetime of a \(\Lambda^{0}\) ) in its own rest frame, and also assuming that the final antiproton is at rest in the lab.