What is the distance that the 217AC will cover before it decays relative to the lab?

Calculating the Distance Traveled by 217AC

Time Dilation and Length Contraction:

To calculate the distance that the 217AC will cover before it decays relative to the lab, we need to take into account the relativistic effects of time dilation and length contraction.

Time Dilation Factor (γ):

The velocity of the 217AC atoms is given as 0.9c, where c is the speed of light. Using the formula γ = 1 / sqrt(1 - (v^2 / c^2)), we can calculate γ as follows:

γ = 1 / sqrt(1 - (0.9c)^2 / c^2)

γ = 1 / sqrt(1 - 0.81)

γ = 1 / sqrt(0.19)

γ ≈ 1 / 0.4359 ≈ 2.296

Length Contraction Factor (β):

Using the formula β = sqrt(1 - (v^2 / c^2)), we can calculate β as follows:

β = sqrt(1 - (0.9c)^2 / c^2)

β = sqrt(1 - 0.81)

β = sqrt(0.19)

β ≈ 0.4359

Calculating Distance Traveled:

The time measured in the lab frame is the rest half-life of the 217 actinium, which is given as 69 ns (69x10^-9 s).

Using the formula d = v * t, where d is the distance, v is the velocity of the particle, and t is the time measured in the lab frame, we can calculate the distance as follows:

d = (0.9c) * (69x10^-9 s)

d = 0.9 * 3x10^8 m/s * 69x10^-9 s ≈ 18.81 meters

Therefore, the distance that the 217AC will cover before it decays relative to the lab is approximately 18.81 meters.