The π-mesons live approximately 8.69 x 10^-8 seconds as viewed in the laboratory, compared to their lifetime of 2.60 x 10^-8 seconds at rest. This means their time appears to slow down due to their high velocity.
To determine the lifetime of the π-mesons as viewed in the laboratory, we need to account for time dilation due to their high velocity. Here's how to solve the problem:
1. Identify the relevant information:
Speed of the π-mesons (v) = 2.70 x 10^8 m/s
Lifetime at rest (τ_0) = 2.60 x 10^-8 s
Speed of light (c) = 3.00 x 10^8 m/s
2. Apply the time dilation formula:
The time dilation formula relates the time measured in the rest frame (τ_0) of the moving object to the time measured in a different frame (τ) by a stationary observer:
τ = τ_0 / √(1 - v^2/c^2)
3. Plug in the values:
τ = (2.60 x 10^-8 s) / √(1 - (2.70 x 10^8 m/s)^2 / (3.00 x 10^8 m/s)^2)
4. Calculate the result:
τ ≈ 8.69 x 10^-8 s
Therefore, the π-mesons live approximately 8.69 x 10^-8 seconds as viewed in the laboratory, compared to their lifetime of 2.60 x 10^-8 seconds at rest. This means their time appears to slow down due to their high velocity.
Bonus: If you need the answer in terms of a multiple of the lifetime at rest, simply divide the calculated lifetime by the rest lifetime:
τ / τ_0 ≈ 8.69 x 10^-8 s / 2.60 x 10^-8 s ≈ 3.34
This means the π-mesons live about 3.34 times longer in the laboratory frame than in their own rest frame.
The probable question is attached in the image.