ions have a half life of 1.8*10exp(-8)sec. A pion beam leaves an accelerator at a speed of 0.8c, What is the expect distance over which half the pion should decay:
(a) classically
(b) relativistically
(a) V*T
where T is the half life and V = 0.8c
(b) V*T'
where T' = T/sqrt[1 - (v/c)^2]= (5/3)T
is the relativistically extended half-lite.
Please refer to the subject as physics, not physic. In English, a physic is a medication.
To determine the expected distance over which half the pions should decay, we can use the concept of the mean lifetime and the velocity of the pions. Let's calculate the values using both classical and relativistic approaches:
(a) Classical Approach:
In classical physics, we can determine the expected distance traveled by considering the concept of the mean lifetime. The mean lifetime of a particle is given by the formula:
mean lifetime (t) = 0.693 * half-life (t₁/₂)
In this case, the half-life of the pions is given as 1.8 * 10^(-8) seconds. We can calculate the mean lifetime as follows:
t = 0.693 * (1.8 * 10^(-8) seconds)
t ≈ 1.24 * 10^(-8) seconds
Since the velocity of the pions is given as 0.8c, where c is the speed of light, we can use the formula:
distance (d) = velocity (v) * time (t)
To find the distance over which half the pions should decay, we substitute the values into the formula:
d = 0.8c * (1.24 * 10^(-8) seconds)
d ≈ 9.92 * 10^(-9) meters
Therefore, the expected distance over which half the pions should decay, using the classical approach, is approximately 9.92 * 10^(-9) meters.
(b) Relativistic Approach:
In the relativistic approach, we need to consider time dilation. The time dilation factor is given by:
time dilation factor (γ) = 1 / √(1 - (v^2 / c^2))
In this case, the velocity of the pions is given as 0.8c, so we can substitute this value into the formula:
γ = 1 / √(1 - (0.8c)^2 / c^2)
γ ≈ 1.67
Now, we need to multiply the mean lifetime (t) by the time dilation factor (γ) to account for the relativistic effects:
t_rel = γ * t
t_rel ≈ 2.07 * 10^(-8) seconds
Finally, we can use the formula for distance:
d_rel = 0.8c * (2.07 * 10^(-8) seconds)
d_rel ≈ 1.66 * 10^(-8) meters
Therefore, the expected distance over which half the pions should decay, using the relativistic approach, is approximately 1.66 * 10^(-8) meters.