A neutral pion has a rest energy of 135 MeV and a mean life of$\mathbf{8}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{17}}\mathbf{s}$ . If it is produced with an initial kinetic energy of 80 MeV and decays after one mean lifetime, what is the longest possible track this particle could leave in a bubble chamber? Use relativistic time dilation.

Initial kinetic energy,$K=80 \text{MeV}$

Rest energy of the neutron,$m{c}^2=135 \mathrm{MeV}$

Mean life of the neutron is, $\tau =8.3\times {10}^{-17} \text{s}$

The expression for the Lorentz factor is given by,

$\gamma =1+\frac{K}{m{c}^2}$

Here role="math" localid="1663156598914" $\gamma$is the Lorentz factor, K is the kinetic energy, $m{c}^2$is rest energy of the neutron.

Using the expression of Lorentz factor,

$\gamma =1+\frac{K}{m{c}^2}$……. (1)

Substitute 80 MeV forK and 135 MeV for$m{c}^2$ in the above equation,

$\begin{array}{rcl}\gamma & =& 1+\frac{80 MeV}{135 MeV}\\ & =& 1+0.59\\ & =& 1.59\end{array}$

The expression for the Lorentz factor in terms of speed is given by,

$\gamma =\frac{1}{\sqrt{1-{\left(\frac{v}{c}\right)}^2}}$……. (ii)

Substitute 1.59 for $\gamma$and$2.99\times {10}^8 \text{m/s}$ for c into the equation (ii)

$\begin{array}{rcl}1.59& =& \frac{1}{\sqrt{1-{\left(\frac{v}{2.99\times {10}^8}\right)}^2}}\\ v& =& 2.33\times {10}^8 \text{m/s}\end{array}$

Life time pion in the laboratory frame is,

$t=\gamma \tau$…… (iii)

Substitute 1.59 for$\gamma$ and $8.3\times {10}^{-17} \text{s}$for$\tau$ into the equation (iii)

$\begin{array}{rcl}t& =& \left(1.59\right)\left(8.3\times {10}^{-17}\right)\\ & =& 1.3\times {10}^{-16}s\end{array}$

The expression for the longest possible track for this particle leaves in a bubble chamber is given by,

$x=vt$….. (iv)

Substitute $2.33\times {10}^8 \text{m/s}$for v and$1.3\times {10}^{-16}\mathrm{s}$ for t into the equation (iv)

$\begin{array}{rcl}x& =& \left(2.33\times {10}^8\right)\left(1.3\times {10}^{-16}\right)\\ & =& 3.1\times {10}^{-8}\mathrm{m}\end{array}$

Therefore the longest possible track for this particle leaves in a bubble chamber is $\begin{array}{rcl}& & 3.1\times {10}^{-8}\mathrm{m}\end{array}$.