Question

Problem 1. In the distant future, a physics student is rushing to her final exam traveling in a spaceship at relativistic speed. She passes a policeman at rest a distance 9 km from a traffic signal (measured in his rest frame).

a) The policeman monitors the student’s tail lights (red light with λ=650 nm in her rest frame) with a spectroscope, viewing directly along her motion away from him. He observes a wavelength λ=1300 nm (infrared). What is her speed relative to the policeman?

b) How long, according to her, does it take to reach the traffic signal?

c) When the policeman stops her for running a red light she claims that she saw the traffic signal as green (i.e., λ=520 nm). Compute the wavelength she would have seen. Is she telling the truth?

Answer 1

Solution:

A)

changein wavelength due to motion of the object is given by the red shift equation

Where = v/c

v is the relative speed between the observer and the source.

Here,

= 650 nm

= 1300 nm

or 1.8×10^8 m/s

b) The actual length is 9 km = 9000 m

The time taken, according to the police officer will be

time = distance/speed = 9000*5/(3*3*10^8) = 5.0*10^-5 s

The time to a moving observer is given by Lorentz transformation equation

Here,

So,

c) Using the equation

Here, velocity should be taken as negative, since the direction of motion changes.

So, = -3/5

Here, for a red signal light,

= 650 nm

So,

= = 0.5*650 = 325 nm, which will come in the UV region.

So, she could not have seen the light as green