A proper lifetime of a certain particle is 100.0ns. a) How long does it live in the lab frame if it moves at
V=0.960c? b) How far does it travel in the lab during that time? c) What is the distance travelled in the
lab frame according to an observer moving with this particle.
To answer these questions, we will use the concept of time dilation and length contraction in special relativity.
a) How long does the particle live in the lab frame if it moves at V=0.960c?
In special relativity, time dilation occurs when an object moves at a significant fraction of the speed of light. The proper lifetime, given as 100.0 ns (nanoseconds), refers to the time experienced by the particle itself.
To find the time experienced in the lab frame, we can use the time dilation formula:
t_lab = t_proper / γ
where γ (gamma) is the Lorentz factor given by:
γ = 1 / sqrt(1 - (V^2 / c^2))
In this case, V = 0.960c, where c is the speed of light. Plugging in the values:
γ = 1 / sqrt(1 - (0.960^2))
Now calculate γ and then find t_lab using the time dilation formula.
b) How far does it travel in the lab during that time?
To find the distance traveled in the lab frame, we can calculate it using the formula:
d = V * t_lab
where d is the distance and V is the velocity of the particle.
c) What is the distance traveled in the lab frame according to an observer moving with this particle?
According to an observer moving with the particle, the distance traveled in the lab frame will be zero. This is because the observer moves along with the particle, and therefore the particle appears stationary from their perspective.
So, to summarize:
a) Calculate γ using the Lorentz factor formula, then find t_lab using the time dilation formula.
b) Calculate the distance d using the formula d = V * t_lab.
c) The distance traveled in the lab frame according to an observer moving with the particle is zero.