Special Relativity: Calculating the Lifetime of an Unstable Particle

How long did the unstable particle live before decaying?

Given that an unstable particle produced in an accelerator experiment travels at a constant velocity, covering 1.00 m in 3.44 ns in the lab frame before changing (decaying) into other particles, how long did it live before decaying?

Answer:

The unstable particle lived for approximately 1.3922 x 10^-8 s before decaying.

In this physics question, the application of Special Relativity and time dilation gives us the lifetime of an unstable particle in its rest frame. After calculating the particle's velocity and the Lorentz factor, we find that the particle lived for about 1.3922 x 10^-8 s in its rest frame before decaying.

Explanation:

The scenario you mentioned pertains to the subject of Special Relativity, specifically time dilation. The time dilation formula is given by: Δt = γΔt₀, where Δt is the time interval observed in the lab frame (frame of reference), γ is the Lorentz factor, and Δt₀ is the proper time interval or the time interval in the particle's rest frame (which we need to find).

To find the rest frame lifetime of the particle, we first need to find γ. We know that γ = 1/√(1 - v²/c²), where v is the velocity of the particle and c is the speed of light. From the question, we are given that the particle travels 1.00 meters in 3.44 ns, so its velocity v = distance/time = 1.00 m / 3.44 ns = 0.29c approximately (where c = 3×10⁸ m/s is the speed of light). Therefore γ = 1.05.

So, applying the time dilation formula, we have Δt₀ = Δt / γ = (3.44 ns) / 1.05 = 1.3922 x 10^-8 s approximately. Therefore, the particle lived for about 1.3922 x 10^-8 s in its rest frame before decaying.