SPECIAL RELATIVITY - PROBLEMS 1
  TOPICS  |  RELATIVITY
[PROBLEMS 2]  [USEFUL EQUATIONS]
TIME DILATION
1) From considering the consservation of energy, calculate the Escape Velocity from the Earth's surface,giving your answer in m/sec.
(Take radius of the earth to be 6400km and mass to be 6x1024 kg.) Now calculate the speed of a spacecraft whose clock runs 1 second slow per hour due to TIME DILATION compared with a clock on Earth. If the fastest speed encountered by Apollo 11 astronauts was achieved during the take-off phase, comment on how the on-board clocks would have differed from those at Mission Control.
 
  2) The Lorentz Equations relate spacetime coordinates in one frame (x,y,z,t) to those in a second frame (x',y',z',t') moving at a steady speed v relative to the first:


Derive the Time Dilation formula from these equations. 3) Two observers A on Earth and B in a spacecraft whose speed is 2 x 108 m / sec both set their watches to the same time.How much time must elapse by A's reckoning before the watches differ by one second?

RELATIVISTIC DOPPLER EFFECT
4)An observer is moving away from a light source at relative speed v. (i) Show that the time between succesive waves arriving is given by T = t +(vt / c) if v is the observer's speed.

(ii) Hence show that

        where to is the 'proper time'
 
 

(iii)Thus show that the observed frequency is related to the source frequency by the formula


This is the Special Relativistic Doppler Frequency shift formula.
UNIVERSAL EXPANSION
  5) A distant galaxy in the constellation Hydra is receding from the Earth at
6.12 x 107 m/sec.
(i) Calculate by how much a green spectral line of wavelength 500nm emitted by this galaxy has been shifted to the red end of the spectrum.

(ii)Calculate also the apparent wavelength of the spectral line.

What colour would it appear to be ?

  6) Plot a graph of the data for these galaxies:



(i) Calculate the ' Recession speed per million light years '
 
 

(ii) If the principal lines in the Hydrogen emission spectrum are Ha =656.3 nm and HB = 486.3 nm then calculate the wavelegths of the two lines for a galaxy that is 1.7x109 light years away and state where in the spectrum the lines would lie.

(iii)A galaxy has a recession speed of 1.92x104 km/sec.An astronomer measures the Haline at 802.4 nm . Comment on this with relevant calculations.
 

 7)A spectral line whose wavelength in a laboratory rest frame is found to be 400nm is shifted to 600 nm in the spectrum of a distant galaxy. Calculate the speed of recession of the galaxy and using the graph from Q6 calculate how far away it is.
 
 

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