An unstable particle is produced in a high$-$energy collision in a particle accelerator. The particle travels a distance of $2\mathrm{.}4cm$ at a speed of $0\mathrm{.}85c,$ both measured relative to the accelerator, before it decays. Calculate the proper lifetime of the particle.

Charged pions are particles present in cosmic ray bursts in Earth's atmosphere. The mean lifetime of a charged pion is $2\mathrm{.}60{10}^{-8}s,$ in a reference frame in

which the pion is at rest. The mean lifetime of charged pions in a particular cosmic ray burst is estimated as $1\mathrm{.}00{10}^{-7}s$ in the reference frame of Earth.

Calculate:

a$.$ the speed of the charged pions in the cosmic ray burst

b$.$ the mean distance travelled by the charged pions relative to Earth.