Question

A beam of particles travels at a speed of 2.4×10⁸m/s. Scientists in the laboratory measure the average lifetime of the particle in the beam as 3.7×10^−6s. Calculate the average lifetime of the particles when they are at rest.

Answer 1

The average lifetime of the particles when they are at rest is approximately 3.5 x 10^-6 seconds.

Step-by-step explanation:

To calculate the average lifetime of particles when they are at rest, we can use the concept of time dilation. Time dilation occurs when an object moves at a high velocity relative to an observer, causing time to appear to slow down for that object.

Using the equation for time dilation, Δt' = (Δt) / γ, where Δt' is the dilated time, Δt is the proper time (or average lifetime of the particle measured in the laboratory), and γ is the Lorentz factor.

In this case, Δt' = 3.7 x 10^-6 s and γ = 1 / sqrt(1 - (v^2 / c^2)), where v is the velocity of the particle and c is the speed of light. Therefore, to find the average lifetime of the particles when they are at rest, we can plug in the values and solve for Δt.

Let's first calculate the value of γ:

γ = 1 / sqrt(1 - (v^2 / c^2))

γ = 1 / sqrt(1 - ((2.4 x 10^8)^2 / (2.7 x 10^8)^2))

γ ≈ 1.056

Now we can calculate Δt:

Δt' = (Δt) / γ

3.7 x 10^-6 s = (Δt) / 1.056

Δt ≈ 3.5 x 10^-6 s

Therefore, the average lifetime of the particles when they are at rest is approximately 3.5 x 10^-6 seconds.