Question

In: Physics

1. Suppose that Alice is rushing past Bob at a velocity u, and she carries with...

1. Suppose that Alice is rushing past Bob at a velocity u, and she carries with her a block of transparent material with index of refraction n. If she shines light through that material then she sees the light moving through it with a velocity c/n. What speed does Bob see the light move through that material? Check that your result has the correct limiting behavior that when n = 1, and u << c/n!

2. An unstable particle of mass M sits at rest in the laboratory. It decays into two identical particles each of mass m which go flying out in opposite directions. What is the velocity of the decay particles, as measured in the lab frame? Hint: you can check that you get the right limit when the mass m goes to zero!

Expert Solution

(1)

Speed of light in a transparent material of refractive index n in Alice frame is

Alice is moving relative to Bob at speed u.

Using Einstein's velocity addition rule, the velocity of light in the transparent material in Bob's frame is

Using v=c/n

For n=1, the speed of light in vacuum in Alice frame is c. We will check whether the speed of light in vaccum is c in Bob's frame also. Substituting n=1 in above relation, we get

The speed is equal to the speed of light in Bob's frame. Our relation behave correctly in this limit.

For u << c/n, when the speed of Bob's frame is very small relative to the speed of light in the material, u can be neglected when addition to c. This is the non-relativistic case, we expect the speed of light in Bob's frame to be c/n.

When Alice is moving at non-relativistic speed, the speed of light in the transparent medium is equal in both frames.

(2)

Initially mass M is at rest, the total energy of the system is equal to the rest mass energy of the particle

The velocity of the daughter particles is equal in magnitude and opposite in direction. Let v₁=v₂=v be the speed of the two daughter particles. The total energy of the daughter particles is

By the conservation energy

When the two identical particles are photons. We know that mass of the photons is zero. Substituting m=0

The speed of the daughter particles is c. We know that photons travel at speed c, therefore our relation gives correct result in limit m goes to zero.

genius_generous answered 2 years ago

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