Chapter 20, Problem 2SP

(a)

To determine

The factor $\gamma$ for velocity of pions relative to the laboratory.

Answer to Problem 2SP

The factor $\gamma$ for velocity of pions relative to the laboratory is $3.203$.

Explanation of Solution

Given info: The velocity of the pions with respect to the laboratory is $0.95c$.

Write the expression to find the $\gamma$ factor.

$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Here,

$v$ is the velocity of pions with respect to laboratory

$c$ is the speed of light

Substitute $0.96c$ for $v$ in the above equation to find $\gamma$.

$\begin{array}{c}\gamma =\frac{1}{\sqrt{1-\frac{{\left(0.95c\right)}^2}{c^2}}}\\ =3.203\end{array}$

Conclusion:

Therefore, the factor $\gamma$ for velocity of pions relative to the laboratory is $3.203$.

(b)

To determine

The half-life of the moving pions as seen by an observer in laboratory.

Answer to Problem 2SP

The half-life of the moving pions as seen by an observer in laboratory is $5.67\times {10}^{-8}\text{ s}$.

Explanation of Solution

Given info: The decay of pions at rest has a half-life of $1.77\times {10}^{-8}\text{ s}$.

Write the expression to find the half-life of the moving pions as seen by the observer in laboratory.

$\begin{array}{c}t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}\\ =t_0\gamma \end{array}$

Here,

$t_0$ is the half-life of pions at rest

$t$ is the half-life of the pions with respect to the observer

Substitute $1.77\times {10}^{-8}\text{ s}$ for $t_0$ and $3.203$ for $\gamma$ in the above equation to find $t$.

$\begin{array}{c}t=\left(1.77\times {10}^{-8}\text{ s}\right)\left(3.023\right)\\ =5.67\times {10}^{-8}\text{ s}\end{array}$

Conclusion:

Therefore, the half-life of the moving pions as seen by an observer in laboratory is $5.67\times {10}^{-8}\text{ s}$.

(c)

To determine

The distance travelled by the pions before its half decay as measured in the laboratory.

Answer to Problem 2SP

The distance travelled by the pions before its half decay as measured in the laboratory is $16.2\text{ m}$.

Explanation of Solution

Given info: The velocity of the pions with respect to the laboratory is $0.95c$.

Write the expression to find the distance travelled by the pion before it has decayed.

$d=vt$

Here,

$d$ is the distance travelled by the pion before it has decayed as measured in the laboratory

Substitute $0.95c$ for $v$ an $5.67\times {10}^{-8}\text{ s}$ for $t$ in the above equation to find $d$.

$\begin{array}{c}d=\left(0.95c\right)\left(5.67\times {10}^{-8}\text{ s}\right)\\ =16.2\text{m}\end{array}$

Conclusion:

Therefore, the distance travelled by the pions before its half decay is $16.2\text{ m}$.

(d)

To determine

The distance travelled by the pions before the half decay from the reference frame of the pion.

Answer to Problem 2SP

The distance travelled from the reference frame of the pion is $5.05\text{ m}$.

Explanation of Solution

Write the expression to find the distance travelled from the reference frame of the pion.

$\begin{array}{c}{d}_0=d\sqrt{1-\frac{v^2}{c^2}}\\ =\frac{d}{\gamma }\end{array}$

Here,

$d_0$ is the distance travelled by the pions before its half decay in the reference frame of the pions

Substitute $16.2\text{ m}$ for $d$ and $3.023$ for $\gamma$ in the above equation to find $d_0$.

$\begin{array}{c}d=\frac{16.2\text{ m}}{3.023}\\ =5.05\text{ m}\end{array}$

Conclusion:

Therefore, the distance travelled before the half decay from the reference frame of the pion is $5.05\text{ m}$.