Particles called mesons are produced by accelerator beams. if these particles travel at 2.70* 10^8 m/s and live 2.60*10^-8 s when at rest relative to an observee how long do they live as viewed in the laboratory?
To determine the lifetime of mesons as viewed in the laboratory, we need to consider time dilation. Time dilation occurs when an object moves relative to an observer, causing time to pass more slowly for the moving object compared to the stationary object.
The formula for time dilation is given by:
Δt' = Δt / γ
Where:
Δt' is the time measured in the laboratory,
Δt is the proper time (time at rest relative to an observer),
and γ is the Lorentz factor, which is defined as γ = 1 / √(1 - v^2 / c^2).
Given:
v = 2.70 * 10^8 m/s (velocity of mesons)
Δt = 2.60 * 10^-8 s (proper time)
Let's calculate the Lorentz factor (γ):
γ = 1 / √(1 - (v^2 / c^2))
= 1 / √(1 - ((2.70 * 10^8)^2 / (3.00 * 10^8)^2))
≈ 1 / 0.196116
≈ 5.099
Now, we can calculate the time measured in the laboratory (Δt'):
Δt' = Δt / γ
= 2.60 * 10^-8 s / 5.099
≈ 5.10 * 10^-9 s
Therefore, the mesons live approximately 5.10 * 10^-9 seconds as viewed in the laboratory.
To calculate the proper lifetime of mesons as viewed in the laboratory, we need to take into account a concept from Special Relativity called time dilation. Time dilation states that the time experienced by a moving object will be dilated or stretched out relative to an observer at rest.
First, let's define the given values:
Speed of the mesons (v) = 2.70 x 10^8 m/s
Rest lifetime of the mesons (τ0) = 2.60 x 10^-8 s
The formula to calculate the proper lifetime (τ) as viewed in the laboratory is:
τ = γ * τ0
where γ (gamma) is the Lorentz factor given by:
γ = 1 / sqrt(1 - (v^2 / c^2))
In this equation, c represents the speed of light in a vacuum and is approximately 3 x 10^8 m/s.
Let's substitute the values into the equations:
γ = 1 / sqrt(1 - ((2.70 x 10^8 m/s)^2 / (3 x 10^8 m/s)^2))
≈ 1 / sqrt(1 - 0.729)
≈ 1 / sqrt(0.271)
≈ 1 / 0.521
≈ 1.920
Now, let's calculate the proper lifetime:
τ = γ * τ0
≈ 1.920 * (2.60 x 10^-8 s)
≈ 4.99 x 10^-8 s
Therefore, as viewed in the laboratory, the mesons would have a proper lifetime of approximately 4.99 x 10^-8 seconds.