In its own frame the average pi meson lives 2.60 x 10^-8s. Suppose it moves at 0.985c relative to the Earth. According to us, on Earth, what is a pi meson's average lifetime? (b) according to us, what is the pi meson;s average travel distance before decaying? (c) If time dilation did not occur what would we see as the pi meson's average travel distance before decaying?
The π meson’s lifetime in its own frame is the proper time
interval, tₒ =2.6•10^−8 s. An earthbound observer measures a longer dilated time interval t . If β =0. 985,
t = tₒ/sqrt(1-β²) = 2.6•10^−8 /sqrt(1-0.985²) =1.5•10^-7 s.
The distance the π meson travels in the earthbound observer’s reference frame is the π-meson’s velocity multiplied by the time interval measured by the earthbound observer. (Note that this is the relative velocity)
d = v•t =0.985•c•1.598510^-7 = 0.985•3•10^8•1.5•10^-7 = 44.5 m,
Without time dilation, the distance traveled would just be the proper lifetime multiplied by the meson’s velocity:
s = 0.985• c•2.6•10^−8 =7.68 m.
To answer these questions, we need to take into account the concept of time dilation, as described by the theory of special relativity.
Time dilation occurs when an object is moving at relativistic speeds, such as the pi meson traveling at 0.985c relative to Earth. The theory states that time appears to move slower for the moving particle compared to an observer at rest.
(a) According to us, on Earth, what is a pi meson's average lifetime?
To find the pi meson's average lifetime as observed from Earth, we need to account for time dilation. The formula for time dilation is:
t' = t / γ
where t' is the observed time, t is the proper time (time experienced by the moving object), and γ is the Lorentz factor given by:
γ = 1 / sqrt(1 - (v^2 / c^2))
Here, v is the velocity of the pi meson relative to Earth (0.985c) and c is the speed of light.
Therefore, we have:
γ = 1 / sqrt(1 - (0.985c)^2 / c^2)
γ = 1 / sqrt(1 - 0.970225)
γ ≈ 3.1324
We know that the proper time of the pi meson is 2.60 x 10^-8s (given in its own frame). So, plugging these values into the time dilation equation:
t' = (2.60 x 10^-8s) / 3.1324
t' ≈ 8.300 x 10^-9s
So, according to us on Earth, the pi meson's average lifetime is approximately 8.300 x 10^-9 seconds.
(b) According to us, what is the pi meson's average travel distance before decaying?
To find the average travel distance, we can use the equation:
d = v * t'
where d is the travel distance, v is the velocity of the pi meson (0.985c), and t' is the observed time found in the previous step.
Plugging in the values:
d = (0.985c) * (8.300 x 10^-9s)
d ≈ 2.565 x 10^-8 meters
So, according to us, the pi meson's average travel distance before decaying is approximately 2.565 x 10^-8 meters.
(c) If time dilation did not occur, what would we see as the pi meson's average travel distance before decaying?
If time dilation did not occur, we would observe the pi meson's proper time instead of the time dilation formula. Therefore, the average travel distance would be calculated using the proper time and velocity:
d = v * t
where t is the proper time (2.60 x 10^-8s) and v is the velocity (0.985c).
Plugging in the values:
d = (0.985c) * (2.60 x 10^-8s)
d ≈ 2.562 x 10^-8 meters
So, if time dilation did not occur, we would see the pi meson's average travel distance before decaying as approximately 2.562 x 10^-8 meters.