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 0.995c, and an observer who is stationary in a laboratory measures the pion's lifetime to be 3.3 × 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

Speed of the pion be v = 0.995 c

A hypothetical person is riding along with the pion.

So both the pion and the hypothetical person are moving with same speed in same direction.

So pion will be at rest in a frame of reference attached to this hypothetical person

Here we are dealing with the concept of time dilation.

Time dilation refers to the fact that clocks moving at speeds close to the speed of the light run slow.

If a person moving along with this clock measures a time interval between any two events using this clock , it is called proper time interval for him

Similarly time interval measured by the hypothetical person is proper time interval for the decay of the pion as both are moving at same speed comparable to the speed of the light in the same direction

Automatically, a person outside this frame observes that the decay of pion takes longer time than it is for the hypothetical person.

Here it is the person in the laboratory

This dilated time interval t' = 3.3 × 10^-8 s

Dilated time interval is related to the proper time interval by the equation

t' = t / ( 1- v²/c² )

Where t is proper time interval

t = t' × ( 1 - v² / c² )

= 3.3 × 10^-8 × ( 1 - 0.995² )

= 3.3 × 10^-8 × 1 - 0.990025

= 3.3 × 10^-8 × 0.009975

= 3.3 × 10^-8 × 0.09987

= 3.2 × 10^-9 s

So the hypothetical person finds that pion decays in 3.2 nanoseconds.

b) Think that both the pion and hypothetical person are at rest and laboratory is moving with a speed of 0.995 c relative to them

we can simply multiply the speed of laboratory relative to the pion ( or the hypothetical person ) with the proper time interval to get the distance traveled by the laboratory before the pions break apart

Distance traveled by the laboratory x = v × t

= 0.995 × 3 × 10⁸ × 3.2 × 10^-9

= 0.955 m

So laboratory travels a distance 0.955 m before the pion breaks apart.