A subatomic particle has a 410ns lifetime in its own rest frame.
If it moves through the lab at 0.970 c, how far does it travel before decaying, as measured in the lab?
To calculate the distance traveled by the subatomic particle before decaying, we need to determine the time it takes for the particle to decay in the lab frame.
Let's start by finding the time dilation factor (γ) for the particle as it moves through the lab at 0.970c. The time dilation factor is given by the equation:
γ = 1 / sqrt(1 - v^2/c^2),
where v is the velocity of the particle and c is the speed of light (3 x 10^8 m/s).
Plugging in the values, we have:
γ = 1 / sqrt(1 - (0.970c)^2/c^2)
= 1 / sqrt(1 - 0.970^2)
≈ 2.906.
This means that time in the lab frame appears to pass approximately 2.906 times slower for the moving particle.
Next, we can calculate the time it takes for the particle to decay in the lab frame. The lifetime of the particle in its own rest frame is given as 410 ns (nanoseconds).
In the lab frame, the decay time (t_lab) will be related to the rest frame decay time (t_rest) by the equation:
t_lab = γ * t_rest.
Substituting the values, we get:
t_lab ≈ 2.906 * 410 ns
≈ 1189.66 ns.
Therefore, the particle takes approximately 1189.66 ns to decay in the lab frame.
Finally, to find the distance traveled (d) by the particle in the lab frame, we can use the equation:
d = v * t_lab,
where v is the velocity of the particle.
Plugging in the values:
d ≈ (0.970c) * 1189.66 ns
≈ 3 x 10^8 m/s * 1189.66 ns
≈ 3.57 x 10^8 meters.
Therefore, the subatomic particle would travel approximately 3.57 x 10^8 meters before decaying, as measured in the lab frame.