Thai is traveling with relativistic speed with respect to the earth. He is heade

Question

Thai is traveling with relativistic speed with respect to the earth. He is headed for Cignus-X to study a black hole up close. To pass the time until he gets there, he is doing an experiment in the lab on his ship. He is creating subatomic particles that are at rest in ship and have a lifetime (from creation to decay) of 3.45 times 10^-6 s. Thuy has stayed behind on earth. Observers in Thuy's frame who record the creation and decay of the particle are 5.000m apart as shown below. When the particle is created: When the particle decays (dies): a) Find v/c for the ship. b) According to Thai on the ship, what is the lifetime of the particle and what is the distance between the two observers in Thuy's frame? c) According to Thuy on earth, what is the lifetime of the particle and what is the distance between the same two observers?

Explanation / Answer

Let's find the answer of (b) first and then calculate (a) v/c and finally (c)

(b) This is clearly given in the question itsef. Acording toThai on the ship ( lab frame of creation and decay of sub-particles), the life time of the particle is 3.45x10^6 s. This is called proper life time.

The distane between the two observers in Thuy's frame is 5000 m as given in the question.

(a) In relativity, the non-proper life time and proper life time are related by

t=t_0/sqrt(1-v^2/c^2) --------(1) t_0=3.45x10^(-6) s

This non-proper time t is measured in Thui's frame from earth.

Distance measured in Thui's frame is d_0=vt=5000 m --------- (2)

Solve equation (1) and (2) you will get v/c=0.979

(c) According to Thuy on earth, the lifetime of the particle is

t=t_0/sqrt(1-v^2/c^2) =3.45e-6/sqrt(1-0.979^2)=1.69X10^-5 s.

We can see from the result t > t_0 which dictates time dialation of relativity. A moving clock ticks more slowly than a clock at rest.

According to Thai's frame, distance between the same two observers is d=d_0xsqrt(1-v^2/c^2)=5000 m x sqrt(1-0.979^2)=1019.3 m which is a consequence of lenth contraction of relativity.

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