Question

A particle known as a pion lives for a short time before breaking apart into other particles. Suppose a pion is moving at a speed of 0.986c, and an observer who is stationary in a laboratory measures the pion's lifetime to be 3.4 × 10-8 s. (a) What is the lifetime according to a hypothetical person who is riding along with the pion? (b) According to this hypothetical person, how far does the laboratory move before the pion breaks apart?

Answer 1

The lifetime of a particle, such as a pion, is measured differently by a stationary observer in a laboratory compared to an observer riding along with the particle. The lifetime as measured by the hypothetical person can be calculated using the Lorentz factor. The distance the laboratory moves before the pion breaks apart can be calculated using the speed of the pion and the lifetime according to the hypothetical person.

Step-by-step explanation:

The lifetime of a particle, such as a pion, is measured differently by an observer who is stationary in a laboratory compared to an observer who is riding along with the particle. According to time dilation, the hypothetical person riding with the pion will measure a longer lifetime compared to the stationary observer.

To calculate the lifetime of the pion according to the hypothetical person, we can use the equation: t' = t / γ, where t' is the measured lifetime, t is the lifetime measured in the laboratory, and γ is the Lorentz factor. In this case, the Lorentz factor can be calculated using the equation: γ = 1/√(1 - (v/c)^2), where v is the speed of the pion and c is the speed of light. Substitute the values into the equations to find the lifetime according to the hypothetical person.

1. The lifetime according to the hypothetical person is t' = 3.4 × 10^(-8) s / γ.
2. The distance the laboratory moves before the pion breaks apart can be calculated using the equation: distance = v * t', where v is the speed of the pion according to the laboratory observer and t' is the lifetime according to the hypothetical person.

Answer 2

Answer: (a) t = 20.36.
`10^(-8)`s

(b) d = 60.225 m

Explanation: According to the Special Relativity Theory proposed by Albert Einstein, the laws of physics are the same for all non-accelerating observers and speed of light is the same in vaccum, no matter the speed the pbserver is travelling. with these statements, Einstein realized that space and time interwoven in a continuum called space-time, i.e., events that occur for one observer at the same time, occurs at different times for a second observer.

(a) To determine the lifetime and since the person is riding along:

t =
`\frac{t_(0) }{\sqrt{(1-v^(2) )/(c^(2) ) } }` , in which:

t is time observed in the other time frame

t₀ is time in the observer's own reference

v is velocity of the object

c is the speed of light in vacuum (c=3.
`10^(8)`m/s)

Calculating:

t =
`\frac{3.4.10^(-8) }{\sqrt{(1 - 0.986c^(2) )/(c^(2) ) } }`

t =
`(3.4.10^(-8) )/(√(0.028) )`

t = 20.36.
`10^(-8)`

The lifetime, according to the hypothetical person is 20.36.
`10^(-8)` seconds.

(b) The pion is moving at a speed of v = 0.986c. In 20.36 seconds,

the laboratory moves:

v=
`(d)/(t)`

d = v.t

d = 0.986.3.
`10^(8)`.20.36.
`10^(-8)`

d = 60.225

The lab moved 60.225 meters.