Question

Question was provided via image upload.

Explanation:

Identify the given values and physical context

The problem asks for the time that passes for moving pions (their proper lifetime, \(\Delta t_0\)) instead of the dilated time observed in the laboratory frame, which is given as \(\Delta t = 2.6 \times 10^{-8}\text{ s}\).

Looking at the faint pencil markings in the background of the image:

• There is a speed factor written as \(0.923\) (likely \(v = 0.923c\)).
• There is a calculation showing \(\Delta t = 2.6 \times 10^{-8}\text{ s}\) as the dilated time, and a calculation for the proper time \(\Delta t_0\).

Let's verify the values:

• Dilated time (lab frame): \(\Delta t = 2.6 \times 10^{-8}\text{ s}\)
• Speed of the pions: \(v = 0.923c\)
• Target: Proper time \(\Delta t_0\) (the time that passes in the pions' rest frame)

Apply the time dilation formula

Using the Time Dilation knowledge point

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

Rearranging to solve for the proper time \(\Delta t_0\):

$$ \Delta t_0 = \Delta t \sqrt{1 - \frac{v^2}{c^2}} $$

Calculate the proper time

Using the Time Dilation knowledge point

$$
LATEXBLOCK0
$$

Answer:

\(1.0 \times 10^{-8}\text{ s}\)